💡First, I need to identify the face value of 7 and the local value of 8 in the number 7098060.The face value of a digit is simply the digit itself, so the face value of 7 is 7.Next, I determine the local value of 8. In the number 7098060, the digit 8 is in the thousands place. Therefore, its local value is 8 multiplied by 1,000, which equals 8,000.Now, I calculate the product of the face value of 7 and the local value of 8:7 multiplied by 8,000 equals 56,000.Finally, I find the local value of 6 in the number 7098060. The digit 6 is in the tens place, so its local value is 6 multiplied by 10, which equals 60.Therefore, the local value of 6 in the product of the face value of 7 and the local value of 8 in the number 7098060 is 60.

💡Okay, so I have this problem where I need to find the remainder when the polynomial (5x^6 + 3x^4 - 2x^3 + 7x^2 + 4) is divided by (x^2 + 2x + 1). Hmm, I remember that when dividing polynomials, the remainder has a degree less than the divisor. Since the divisor here is a quadratic polynomial (degree 2), the remainder should be a linear polynomial, which means it will look something like (ax + b), where (a) and (b) are constants that I need to find.Alright, so let me write that down. Let me denote the dividend as (P(x) = 5x^6 + 3x^4 - 2x^3 + 7x^2 + 4) and the divisor as (D(x) = x^2 + 2x + 1). The division algorithm tells me that there exist unique polynomials (Q(x)) (the quotient) and (R(x)) (the remainder) such that:[P(x) = D(x) cdot Q(x) + R(x)]And since (D(x)) is degree 2, (R(x)) must be degree 1 or less, so (R(x) = ax + b).Now, I recall that (D(x) = x^2 + 2x + 1) can be factored as ((x + 1)^2). That might be useful because it has a repeated root at (x = -1). I think I can use this fact to find the remainder by plugging in (x = -1) into the equation (P(x) = D(x) cdot Q(x) + R(x)). Let me try that. If I substitute (x = -1), then (D(-1) = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0). So, plugging (x = -1) into the equation gives:[P(-1) = 0 cdot Q(-1) + R(-1)][P(-1) = R(-1)]Calculating (P(-1)):[P(-1) = 5(-1)^6 + 3(-1)^4 - 2(-1)^3 + 7(-1)^2 + 4][= 5(1) + 3(1) - 2(-1) + 7(1) + 4][= 5 + 3 + 2 + 7 + 4][= 21]So, (R(-1) = 21). But (R(x) = ax + b), so:[R(-1) = a(-1) + b = -a + b = 21]That's one equation: (-a + b = 21). I need another equation to solve for both (a) and (b). I remember that since (D(x)) has a repeated root, I can take the derivative of both sides of the equation (P(x) = D(x) cdot Q(x) + R(x)) and then substitute (x = -1) again. Let me try that. Differentiating both sides:[P'(x) = D'(x) cdot Q(x) + D(x) cdot Q'(x) + R'(x)]But at (x = -1), (D(-1) = 0), so the term (D(x) cdot Q'(x)) becomes zero. Therefore, at (x = -1):[P'(-1) = D'(-1) cdot Q(-1) + R'(-1)]But I don't know (Q(-1)), so maybe I can express it in terms of known quantities. Wait, maybe I can rearrange the equation:[P'(x) = D'(x) cdot Q(x) + D(x) cdot Q'(x) + R'(x)]At (x = -1), (D(-1) = 0), so:[P'(-1) = D'(-1) cdot Q(-1) + R'(-1)]But I don't know (Q(-1)), so maybe I need another approach. Alternatively, perhaps I can use the fact that (R(x)) is linear, so its derivative (R'(x)) is just (a). Let me compute (P'(x)):[P(x) = 5x^6 + 3x^4 - 2x^3 + 7x^2 + 4][P'(x) = 30x^5 + 12x^3 - 6x^2 + 14x]Now, evaluate (P'(-1)):[P'(-1) = 30(-1)^5 + 12(-1)^3 - 6(-1)^2 + 14(-1)][= 30(-1) + 12(-1) - 6(1) + 14(-1)][= -30 - 12 - 6 - 14][= -62]So, (P'(-1) = -62). Now, (D(x) = (x + 1)^2), so (D'(x) = 2(x + 1)). Therefore, (D'(-1) = 2(-1 + 1) = 0). Wait, that's zero again. Hmm, so plugging into the derivative equation:[P'(-1) = D'(-1) cdot Q(-1) + R'(-1)][-62 = 0 cdot Q(-1) + a][a = -62]Wait, but earlier I thought (a = -38). Did I make a mistake? Let me check my calculations again.Wait, no, actually, I think I made a mistake in the derivative calculation. Let me recalculate (P'(-1)):[P'(x) = 30x^5 + 12x^3 - 6x^2 + 14x][P'(-1) = 30(-1)^5 + 12(-1)^3 - 6(-1)^2 + 14(-1)][= 30(-1) + 12(-1) - 6(1) + 14(-1)][= -30 - 12 - 6 - 14][= -62]No, that seems correct. So (a = -62). But earlier, when I thought I had (a = -38), that was probably incorrect. Let me see where I went wrong.Wait, in the initial problem, the user had a different approach where they used the derivative, but I think I misapplied it. Let me try a different method to verify.Alternatively, I can use polynomial long division, but that might be time-consuming. Alternatively, I can express (P(x)) in terms of (D(x)) and find the remainder.Since (D(x) = (x + 1)^2), I can express (P(x)) as:[P(x) = (x + 1)^2 cdot Q(x) + ax + b]As before, plugging (x = -1) gives:[P(-1) = a(-1) + b = -a + b = 21]So, equation 1: (-a + b = 21).Now, to find another equation, I can take the derivative of both sides:[P'(x) = 2(x + 1) cdot Q(x) + (x + 1)^2 cdot Q'(x) + a]At (x = -1), the terms involving (Q(x)) and (Q'(x)) become zero because (x + 1 = 0), so:[P'(-1) = a]We already calculated (P'(-1) = -62), so (a = -62).Now, plugging (a = -62) into equation 1:[-(-62) + b = 21][62 + b = 21][b = 21 - 62][b = -41]Wait, that's different from the initial answer I thought of. So, the remainder should be (-62x - 41). But in the initial problem, the user had a different answer. Let me check my calculations again.Wait, perhaps I made a mistake in calculating (P(-1)). Let me recalculate (P(-1)):[P(-1) = 5(-1)^6 + 3(-1)^4 - 2(-1)^3 + 7(-1)^2 + 4][= 5(1) + 3(1) - 2(-1) + 7(1) + 4][= 5 + 3 + 2 + 7 + 4][= 21]That's correct. So, equation 1 is correct: (-a + b = 21).Then, (P'(-1) = -62), so (a = -62). Therefore, (b = 21 + a = 21 - 62 = -41).So, the remainder is (-62x - 41).Wait, but in the initial problem, the user had a different answer. Let me check their steps.Wait, the user had:They plugged in (x = -1) and got (b - a = 9), which seems incorrect because I got (b - a = 21). So, perhaps they made a mistake in their calculation of (P(-1)). Let me check their calculation:They wrote:(5(-1)^6 + 3(-1)^4 - 2(-1)^3 + 7(-1)^2 + 4)Which is:5(1) + 3(1) - 2(-1) + 7(1) + 4= 5 + 3 + 2 + 7 + 4 = 21But they wrote it equals 9. That's incorrect. So, their initial equation was wrong, leading to wrong values for (a) and (b).Therefore, the correct remainder should be (-62x - 41).But wait, let me double-check by performing polynomial long division or synthetic division to confirm.Alternatively, I can express (P(x)) as ( (x + 1)^2 cdot Q(x) + ax + b ) and expand it, then equate coefficients.But that might be time-consuming. Alternatively, I can use the Remainder Factor Theorem for repeated roots.Since (D(x) = (x + 1)^2), the remainder (R(x)) must satisfy:1. (P(-1) = R(-1))2. (P'(-1) = R'(-1))We already used these two conditions to find (a) and (b).So, with (R(x) = ax + b), we have:1. (R(-1) = -a + b = 21)2. (R'(-1) = a = -62)Thus, (a = -62), and substituting back, (b = 21 + 62 = 83). Wait, that contradicts my earlier result. Wait, no:Wait, (R(-1) = -a + b = 21). If (a = -62), then:(-(-62) + b = 21)(62 + b = 21)(b = 21 - 62 = -41)Yes, that's correct. So, (R(x) = -62x - 41).Wait, but earlier I thought the user had a different answer, but perhaps I confused it. Let me check the initial problem again.Wait, the initial problem was:"Find the remainder when (5x^6 + 3x^4 - 2x^3 + 7x^2 + 4) is divided by (x^2 + 2x + 1)."And the user's solution was:They said (x^2 + 2x + 1 = (x+1)^2), so remainder is (ax + b). Then, they plugged in (x = -1) and got (b - a = 9), which is incorrect because (P(-1) = 21), not 9. Then, they took the derivative and found (a = -38), leading to (b = -29). But that's wrong because their calculation of (P(-1)) was wrong.So, the correct remainder is (-62x - 41).But wait, let me verify this by another method. Let me perform polynomial long division.Dividing (5x^6 + 3x^4 - 2x^3 + 7x^2 + 4) by (x^2 + 2x + 1).First, divide (5x^6) by (x^2) to get (5x^4). Multiply (5x^4) by (x^2 + 2x + 1) to get (5x^6 + 10x^5 + 5x^4). Subtract this from the original polynomial:[(5x^6 + 3x^4 - 2x^3 + 7x^2 + 4) - (5x^6 + 10x^5 + 5x^4) = -10x^5 - 2x^4 - 2x^3 + 7x^2 + 4]Next, divide (-10x^5) by (x^2) to get (-10x^3). Multiply (-10x^3) by (x^2 + 2x + 1) to get (-10x^5 - 20x^4 - 10x^3). Subtract:[(-10x^5 - 2x^4 - 2x^3 + 7x^2 + 4) - (-10x^5 - 20x^4 - 10x^3) = 18x^4 + 8x^3 + 7x^2 + 4]Next, divide (18x^4) by (x^2) to get (18x^2). Multiply (18x^2) by (x^2 + 2x + 1) to get (18x^4 + 36x^3 + 18x^2). Subtract:[(18x^4 + 8x^3 + 7x^2 + 4) - (18x^4 + 36x^3 + 18x^2) = -28x^3 - 11x^2 + 4]Next, divide (-28x^3) by (x^2) to get (-28x). Multiply (-28x) by (x^2 + 2x + 1) to get (-28x^3 - 56x^2 - 28x). Subtract:[(-28x^3 - 11x^2 + 4) - (-28x^3 - 56x^2 - 28x) = 45x^2 + 28x + 4]Next, divide (45x^2) by (x^2) to get (45). Multiply (45) by (x^2 + 2x + 1) to get (45x^2 + 90x + 45). Subtract:[(45x^2 + 28x + 4) - (45x^2 + 90x + 45) = -62x - 41]So, the remainder is (-62x - 41), which matches what I found earlier. Therefore, the correct remainder is (-62x - 41).But wait, in the initial problem, the user had a different answer. Let me check their steps again.They wrote:"Plugging in (x = -1) gives:(5(-1)^6 + 3(-1)^4 - 2(-1)^3 + 7(-1)^2 + 4 = 0 + a(-1) + b),which simplifies to:(b - a = 9)."But as I calculated, (P(-1) = 21), so (b - a = 21), not 9. So, their initial equation was wrong, leading to incorrect values for (a) and (b).Therefore, the correct remainder is (-62x - 41).Wait, but in the initial problem, the user had a different answer, so perhaps I should double-check my calculations once more.Let me recalculate (P(-1)):[P(-1) = 5(-1)^6 + 3(-1)^4 - 2(-1)^3 + 7(-1)^2 + 4][= 5(1) + 3(1) - 2(-1) + 7(1) + 4][= 5 + 3 + 2 + 7 + 4][= 21]Yes, that's correct. So, (b - a = 21).Then, (P'(x) = 30x^5 + 12x^3 - 6x^2 + 14x), so (P'(-1)):[30(-1)^5 + 12(-1)^3 - 6(-1)^2 + 14(-1)][= -30 - 12 - 6 - 14][= -62]So, (a = -62), and (b = 21 + a = 21 - 62 = -41).Thus, the remainder is (-62x - 41).I think that's correct. The initial user's answer was wrong because they miscalculated (P(-1)) as 9 instead of 21.

💡Okay, so I have this problem here where I need to find the value of ( r ) as ( m ) approaches zero. The problem defines ( L(m) ) as the ( x )-coordinate of the left endpoint of the intersection of the graphs ( y = x^2 - 4 ) and ( y = m ), where ( -4 < m < 4 ). Then, ( r ) is defined as ( frac{L(-m) - L(m)}{m} ). I need to determine the value of ( r ) as ( m ) approaches zero.First, I should figure out what ( L(m) ) is. Since ( L(m) ) is the ( x )-coordinate of the left endpoint of the intersection, that means it's the smaller ( x )-value where the two graphs meet. The equation ( y = x^2 - 4 ) is a parabola opening upwards with its vertex at ( (0, -4) ). The line ( y = m ) is a horizontal line. The intersections occur where ( x^2 - 4 = m ), so solving for ( x ), we get ( x^2 = m + 4 ), which means ( x = pm sqrt{m + 4} ). Since we're interested in the left endpoint, that would be the negative root, so ( L(m) = -sqrt{m + 4} ).Similarly, ( L(-m) ) would be the left endpoint when the horizontal line is ( y = -m ). So, following the same logic, ( L(-m) = -sqrt{-m + 4} ).Now, let's plug these into the expression for ( r ):[r = frac{L(-m) - L(m)}{m} = frac{-sqrt{-m + 4} - (-sqrt{m + 4})}{m} = frac{-sqrt{4 - m} + sqrt{4 + m}}{m}]So, ( r = frac{sqrt{4 + m} - sqrt{4 - m}}{m} ).I need to find the limit of ( r ) as ( m ) approaches zero. That is:[lim_{m to 0} frac{sqrt{4 + m} - sqrt{4 - m}}{m}]This looks like a difference quotient, which is typically used to find derivatives. In fact, this resembles the definition of the derivative of the function ( f(x) = sqrt{x} ) at ( x = 4 ). Let me think about that.If I let ( f(x) = sqrt{x} ), then the derivative ( f'(x) ) is ( frac{1}{2sqrt{x}} ). So, ( f'(4) = frac{1}{2sqrt{4}} = frac{1}{4} ). But wait, in our case, the expression is a bit different because we have ( sqrt{4 + m} - sqrt{4 - m} ) over ( m ). Is this related to the derivative?Alternatively, I can rationalize the numerator to simplify the expression. Let me try that.Multiply the numerator and denominator by the conjugate of the numerator:[frac{sqrt{4 + m} - sqrt{4 - m}}{m} times frac{sqrt{4 + m} + sqrt{4 - m}}{sqrt{4 + m} + sqrt{4 - m}} = frac{(4 + m) - (4 - m)}{m(sqrt{4 + m} + sqrt{4 - m})}]Simplifying the numerator:[(4 + m) - (4 - m) = 4 + m - 4 + m = 2m]So now, the expression becomes:[frac{2m}{m(sqrt{4 + m} + sqrt{4 - m})} = frac{2}{sqrt{4 + m} + sqrt{4 - m}}]Now, taking the limit as ( m ) approaches zero:[lim_{m to 0} frac{2}{sqrt{4 + m} + sqrt{4 - m}} = frac{2}{sqrt{4} + sqrt{4}} = frac{2}{2 + 2} = frac{2}{4} = frac{1}{2}]So, the limit is ( frac{1}{2} ).Wait, but earlier I thought about the derivative and got ( frac{1}{4} ). Did I make a mistake there? Let me check.The expression ( frac{sqrt{4 + m} - sqrt{4 - m}}{m} ) can be seen as:[frac{f(4 + m) - f(4 - m)}{m}]where ( f(x) = sqrt{x} ). This is actually the symmetric difference quotient, which is used to approximate the derivative. The symmetric difference quotient is:[frac{f(a + h) - f(a - h)}{2h}]In our case, ( a = 4 ) and ( h = m ), so the expression is:[frac{f(4 + m) - f(4 - m)}{m} = 2 times frac{f(4 + m) - f(4 - m)}{2m}]Which means:[frac{sqrt{4 + m} - sqrt{4 - m}}{m} = 2 times text{symmetric difference quotient}]And as ( m ) approaches zero, the symmetric difference quotient approaches the derivative ( f'(4) ). So:[lim_{m to 0} frac{sqrt{4 + m} - sqrt{4 - m}}{m} = 2 times f'(4) = 2 times frac{1}{4} = frac{1}{2}]So, that confirms the earlier result. The limit is indeed ( frac{1}{2} ).Therefore, the value of ( r ) as ( m ) approaches zero is ( frac{1}{2} ).**Final Answer**boxed{B}

💡Alright, so I have this problem here: It says that A, B, and the square root of A plus the square root of B are all rational numbers. And I need to prove that the square roots of A and B themselves are rational. Hmm, okay, let's see how to approach this.First off, I know that if a number is rational, it can be expressed as a fraction of two integers. So, A and B are both rational, and their square roots add up to another rational number. That seems interesting because square roots can sometimes be irrational, like the square root of 2. But in this case, the sum of these two square roots is rational, which might impose some restrictions on what A and B can be.Let me denote sqrt(A) as x and sqrt(B) as y for simplicity. So, x + y is rational, and A = x², B = y² are also rational. I need to show that x and y are rational.Since x + y is rational, let's call that sum r, where r is a rational number. So, x + y = r. Also, since A and B are rational, x² and y² are rational. Maybe I can use these two pieces of information to find something about x and y.Let me square both sides of the equation x + y = r. That gives me (x + y)² = r², which expands to x² + 2xy + y² = r². I know that x² and y² are rational, so their sum x² + y² is also rational. Therefore, 2xy must be rational as well because r² is rational and x² + y² is rational. So, 2xy is rational, which means that xy is rational.Now I have two equations: x + y = r (rational) and xy = s (rational, since 2xy is rational). So, x and y are roots of the quadratic equation t² - rt + s = 0. The roots of this equation can be found using the quadratic formula: t = [r ± sqrt(r² - 4s)] / 2.For x and y to be rational, the discriminant, which is r² - 4s, must be a perfect square of a rational number. Let's check what r² - 4s is. Since r and s are both rational, r² - 4s is also rational. So, the square root of this discriminant must be rational as well for the roots to be rational.Wait, but how do I know that sqrt(r² - 4s) is rational? I mean, just because r² - 4s is rational doesn't automatically mean its square root is rational. For example, 2 is rational, but sqrt(2) is irrational. So, I need to ensure that r² - 4s is a perfect square of a rational number.Let me think about this. From the earlier equations, I have x + y = r and xy = s. Also, since x² and y² are rational, and x and y are real numbers (because A and B are under square roots and presumably non-negative), I can consider that x and y are either both rational or both irrational.But if x and y were both irrational, their sum could still be rational, like sqrt(2) + (1 - sqrt(2)) = 1, which is rational. However, in this case, their product would be irrational unless they are conjugate surds. But in our case, the product xy = s is rational. So, if x and y are irrational, their product being rational suggests that they might be conjugate surds.But wait, if x and y are conjugate surds, their sum would also be rational, which is consistent with x + y = r. However, in that case, the discriminant r² - 4s would be positive but not a perfect square, leading to irrational roots. But we need the roots to be rational, so this seems contradictory.Hmm, maybe I need to approach this differently. Let's consider that if x and y are both irrational, their sum is rational, and their product is rational. That would mean that x and y are roots of a quadratic equation with rational coefficients, which is what we have: t² - rt + s = 0. The roots of such an equation are either both rational or both irrational conjugates.But in our case, if x and y are irrational conjugates, then they cannot be expressed as rational numbers, which contradicts the requirement that sqrt(A) and sqrt(B) are rational. Therefore, the only possibility is that x and y are both rational.Wait, that makes sense. If x and y were irrational, their sum and product being rational would imply they are conjugate surds, but then they wouldn't be rational. Since we need x and y to be rational, the discriminant must be a perfect square, ensuring that the roots are rational.So, putting it all together: Given that A, B, and sqrt(A) + sqrt(B) are rational, we set x = sqrt(A), y = sqrt(B), and find that x + y and xy are rational. This leads us to the quadratic equation t² - (x + y)t + xy = 0, whose discriminant must be a perfect square for the roots to be rational. Since the discriminant is rational, it must be a perfect square of a rational number, ensuring that x and y are rational.Therefore, sqrt(A) and sqrt(B) must be rational numbers.

💡Okay, so I have this equation here: 16x² - 32x + 16y² + 64y + 64 = 0. I need to find the radius of the circle defined by this equation. Hmm, I remember that the general equation of a circle is something like (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. But this equation doesn't look like that right now. It has coefficients in front of the x² and y² terms, and it's all set equal to zero. So, I think I need to manipulate this equation to get it into the standard circle form.First, I notice that both the x² and y² terms have coefficients of 16. Maybe I can factor out the 16 from those terms to simplify things. Let me try that:16x² - 32x + 16y² + 64y + 64 = 0Factor out 16 from the x terms and the y terms:16(x² - 2x) + 16(y² + 4y) + 64 = 0Hmm, okay, so now I have 16 factored out from both the x and y parts. The next step is probably to complete the square for both the x and y terms. Completing the square is a method used to form a perfect square trinomial, which will help me get the equation into the standard circle form.Let me recall how completing the square works. For any quadratic expression like x² + bx, I can complete the square by adding and subtracting (b/2)². This will give me (x + b/2)² - (b/2)². I need to do this for both the x and y terms in my equation.Starting with the x terms: x² - 2x. Here, b is -2. So, (b/2)² would be (-2/2)² = (-1)² = 1. So, I add and subtract 1 inside the parentheses:16[(x² - 2x + 1 - 1)] + 16(y² + 4y) + 64 = 0This simplifies to:16[(x - 1)² - 1] + 16(y² + 4y) + 64 = 0Now, moving on to the y terms: y² + 4y. Here, b is 4. So, (b/2)² is (4/2)² = 2² = 4. I add and subtract 4 inside the parentheses:16[(x - 1)² - 1] + 16[(y² + 4y + 4 - 4)] + 64 = 0This simplifies to:16[(x - 1)² - 1] + 16[(y + 2)² - 4] + 64 = 0Now, let's distribute the 16 into each bracket:16(x - 1)² - 16*1 + 16(y + 2)² - 16*4 + 64 = 0Calculating the constants:16(x - 1)² - 16 + 16(y + 2)² - 64 + 64 = 0Combine the constants:-16 - 64 + 64 = -16So, the equation becomes:16(x - 1)² + 16(y + 2)² - 16 = 0Now, let's move the constant term to the other side:16(x - 1)² + 16(y + 2)² = 16Hmm, I can factor out the 16 from the left side, but actually, since both terms have 16, maybe I can divide the entire equation by 16 to simplify it further:Dividing each term by 16:(x - 1)² + (y + 2)² = 1Ah, now this looks like the standard equation of a circle. Comparing it to (x - h)² + (y - k)² = r², I can see that the center of the circle is at (h, k) = (1, -2), and the radius squared, r², is equal to 1. Therefore, the radius r is the square root of 1, which is 1.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the original equation, I factored out 16, completed the square for both x and y terms, and then simplified. The constants after completing the square gave me -16, which I moved to the other side, resulting in 16 on the right. Dividing everything by 16 gave me the standard circle equation with r² = 1, so r = 1. Yeah, that seems correct.I think I got it right. The radius is 1.

💡Okay, so I need to figure out how to convert the decimal number 156 into base four. I remember that converting between bases involves dividing the number by the base and keeping track of the remainders. But I'm a bit fuzzy on the exact steps, so I'll have to think it through carefully.First, I know that base four means each digit represents a power of four. The places are 4^0 (which is 1), 4^1 (which is 4), 4^2 (which is 16), 4^3 (which is 64), 4^4 (which is 256), and so on. Since 156 is less than 256, I don't need to go beyond 4^3 for this conversion.So, I think the process is to divide the number by the highest power of four that fits into it, then take the remainder and repeat the process with the next lower power of four. Let me try that.First, what's the highest power of four less than or equal to 156? Well, 4^3 is 64, and 4^4 is 256, which is too big. So, I'll start with 64.I divide 156 by 64. Let's see, 64 times 2 is 128, and 64 times 3 is 192, which is too much. So, 2 times 64 is 128. That means the coefficient for 4^3 is 2.Now, subtract 128 from 156 to find the remainder. 156 minus 128 is 28. Okay, so now I have 28 left to convert.Next, I move to the next lower power of four, which is 4^2, or 16. I divide 28 by 16. 16 times 1 is 16, and 16 times 2 is 32, which is too big. So, the coefficient for 4^2 is 1.Subtract 16 from 28, and I get 12. Now, I have 12 left.Moving down to the next power, which is 4^1, or 4. I divide 12 by 4. 4 times 3 is 12, so the coefficient here is 3.Subtract 12 from 12, and I get 0. There's nothing left, so the coefficient for 4^0 is 0.Putting it all together, the coefficients for each power of four are 2 (for 4^3), 1 (for 4^2), 3 (for 4^1), and 0 (for 4^0). So, the base four representation of 156 in decimal is 2130 in base four.Wait, let me double-check that to make sure I didn't make a mistake. If I convert 2130 base four back to decimal, does it equal 156?Starting from the right, the digits are 0, 3, 1, and 2. So, calculating:0 * 4^0 = 0 * 1 = 03 * 4^1 = 3 * 4 = 121 * 4^2 = 1 * 16 = 162 * 4^3 = 2 * 64 = 128Adding those up: 0 + 12 + 16 + 128 = 156. Yep, that checks out. So, I think I did it right.I guess another way to think about it is by using successive divisions. Let me try that method too to confirm.Starting with 156, divide by 4. 156 divided by 4 is 39 with a remainder of 0. So, the least significant digit is 0.Now, take 39 and divide by 4. 39 divided by 4 is 9 with a remainder of 3. So, the next digit is 3.Take 9 and divide by 4. 9 divided by 4 is 2 with a remainder of 1. So, the next digit is 1.Take 2 and divide by 4. 2 divided by 4 is 0 with a remainder of 2. So, the next digit is 2.Since the quotient is now 0, we stop. Reading the remainders from last to first, we get 2 1 3 0, which is 2130 in base four. That matches what I got earlier. So, I'm confident that's the correct answer.I think I understand the process now. It's about breaking down the number into chunks that fit the powers of the base and keeping track of how many times each power fits into the remaining number. The remainders give the digits of the new base representation, starting from the least significant digit.It's interesting how both methods—direct division and successive division—lead to the same result. I guess it's because they're fundamentally doing the same thing but approached from slightly different angles. The direct method I used first involves identifying the highest power first and working down, while the successive division method works from the lowest power up by repeatedly dividing by the base.Either way, the key is to keep track of the remainders and ensure that each step is accurate. It's also a good idea to verify the result by converting back to the original base to confirm that it matches the starting number. That helps catch any mistakes that might have been made during the conversion process.I wonder if there's a quicker way or a formula to do this without having to manually divide each time. Maybe using logarithms to find the highest power needed? Let me think about that.If I take the logarithm base 4 of 156, it should give me the exponent of the highest power of 4 that fits into 156. So, log base 4 of 156 is equal to log(156)/log(4). Calculating that, log(156) is approximately 2.193, and log(4) is approximately 0.602. So, 2.193 divided by 0.602 is roughly 3.645. That means the highest integer power is 3, which corresponds to 4^3 = 64. That matches what I found earlier.So, using logarithms can help identify the starting point for the highest power needed. That might be useful for larger numbers where manually figuring out the highest power could be time-consuming.Another thing I'm thinking about is how this process scales. For example, if I had a much larger number, say in the thousands or even larger, this method would still work, but it would require more steps. It might be helpful to write down each step clearly to avoid confusion or mistakes.Also, I recall that in some cases, people use positional notation to keep track of the digits as they convert. For instance, writing down the powers of four and seeing how many times each fits into the number. That seems similar to what I did initially.I also wonder about the applications of base four. It's not as commonly used as binary or hexadecimal, but I think it has some niche applications, especially in certain areas of computer science or mathematics where base four is more convenient or natural. Understanding different bases is important because it helps in grasping how numbers are represented and manipulated in various systems.In summary, converting 156 from base ten to base four involves determining how many times each power of four fits into the number, starting from the highest power down to the lowest. By carefully dividing and keeping track of remainders, I arrived at the base four equivalent of 2130. Verifying by converting back to base ten confirmed the accuracy of the result. This exercise helped reinforce my understanding of base conversion and the importance of systematic approaches to avoid errors.