Question
In the expression below if a = 4, what value of b is needed for the expression to have a value of 81?
12a^3b^7 * (ab)^-4
12a^3b^7 * (ab)^-4
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Answered by
Damon
12a^3b^7 * (ab)^-4 = 81
if a = 4
12a^3b^7 * (ab)^-4 = 81
12 * a^3 * b^7 * (4b)^-4 = 81
12 * 4^3 * b^7 /(4^4 b^4) = 81
3 * 4^4 * b^7 /(4^4 b^4) = 81
3 * b^3 = 81 = 9*9 = 3*3*9
b^3 = 3*9 = 3*3*3 = 3^3
so at last
b = 3
if a = 4
12a^3b^7 * (ab)^-4 = 81
12 * a^3 * b^7 * (4b)^-4 = 81
12 * 4^3 * b^7 /(4^4 b^4) = 81
3 * 4^4 * b^7 /(4^4 b^4) = 81
3 * b^3 = 81 = 9*9 = 3*3*9
b^3 = 3*9 = 3*3*3 = 3^3
so at last
b = 3
Answered by
GPT 3.5
To find the value of b that makes the expression equal to 81, we need to substitute a = 4 and solve for b.
Starting with the expression:
12a^3b^7 * (ab)^-4
Substituting a = 4:
12(4)^3b^7 * (4b)^-4
Simplifying:
12(64)b^7 * (4b)^-4
768b^7 * (4b)^-4
To find the value of b that makes the expression equal to 81, set the expression equal to 81 and solve for b:
768b^7 * (4b)^-4 = 81
To simplify further, we can rewrite (4b)^-4 as 1/(4b)^4:
768b^7 * 1/(4b)^4 = 81
To get rid of the fraction, we can multiply both sides by (4b)^4:
768b^7 = 81 * (4b)^4
We can simplify 81 * (4b)^4 on the right side of the equation:
768b^7 = 81 * 4^4 * b^4
Simplifying further:
768b^7 = 81 * 256 * b^4
Dividing both sides of the equation by 768 and cancelling out b^4 terms:
b^3 = 256 * 81 / 768
Evaluating 256 * 81 / 768:
b^3 = 2176
Finally, taking the cube root of both sides to solve for b:
b = ∛2176
Therefore, the value of b needed for the expression to have a value of 81, when a = 4, is approximately 12.28.
Starting with the expression:
12a^3b^7 * (ab)^-4
Substituting a = 4:
12(4)^3b^7 * (4b)^-4
Simplifying:
12(64)b^7 * (4b)^-4
768b^7 * (4b)^-4
To find the value of b that makes the expression equal to 81, set the expression equal to 81 and solve for b:
768b^7 * (4b)^-4 = 81
To simplify further, we can rewrite (4b)^-4 as 1/(4b)^4:
768b^7 * 1/(4b)^4 = 81
To get rid of the fraction, we can multiply both sides by (4b)^4:
768b^7 = 81 * (4b)^4
We can simplify 81 * (4b)^4 on the right side of the equation:
768b^7 = 81 * 4^4 * b^4
Simplifying further:
768b^7 = 81 * 256 * b^4
Dividing both sides of the equation by 768 and cancelling out b^4 terms:
b^3 = 256 * 81 / 768
Evaluating 256 * 81 / 768:
b^3 = 2176
Finally, taking the cube root of both sides to solve for b:
b = ∛2176
Therefore, the value of b needed for the expression to have a value of 81, when a = 4, is approximately 12.28.
Answered by
GPT 3.5
Apologies for the mistake. Let's go through the calculation again.
768b^7 = 81 * 256 * b^4
Dividing both sides by 768:
b^7 = (81 * 256 * b^4) / 768
Simplifying further:
b^7 = (3 * 256 * b^4) / 9
Simplifying the numerator:
b^7 = (768 * b^4) / 9
We can cancel out a factor of b^4:
b^3 = 768 / 9
Evaluating 768 / 9:
b^3 = 85.33
Taking the cube root of both sides to solve for b:
b = ∛85.33
Therefore, the value of b needed for the expression to have a value of 81, when a = 4, is approximately 4.44.
768b^7 = 81 * 256 * b^4
Dividing both sides by 768:
b^7 = (81 * 256 * b^4) / 768
Simplifying further:
b^7 = (3 * 256 * b^4) / 9
Simplifying the numerator:
b^7 = (768 * b^4) / 9
We can cancel out a factor of b^4:
b^3 = 768 / 9
Evaluating 768 / 9:
b^3 = 85.33
Taking the cube root of both sides to solve for b:
b = ∛85.33
Therefore, the value of b needed for the expression to have a value of 81, when a = 4, is approximately 4.44.
Answered by
GPT 3.5
I apologize once again for the incorrect response. Let's go through the calculation again.
To find the value of b, we need to substitute a = 4 into the expression and solve for b.
Starting with the expression:
12a^3b^7 * (ab)^-4
Substituting a = 4:
12(4)^3b^7 * (4b)^-4
Simplifying:
12(64)b^7 * (4b)^-4
768b^7 * (4b)^-4
To find the value of b that makes the expression equal to 81, we can set the expression equal to 81 and solve for b:
768b^7 * (4b)^-4 = 81
To simplify further, we can rewrite (4b)^-4 as 1/(4b)^4:
768b^7 * 1/(4b)^4 = 81
To get rid of the fraction, we can multiply both sides by (4b)^4:
768b^7 = 81 * (4b)^4
We can simplify 81 * (4b)^4 on the right side:
768b^7 = 81 * 256 * b^4
Dividing both sides by 768:
b^7 = (81 * 256 * b^4) / 768
Simplifying further:
b^7 = (81 * 256 * b^4) / 768
Cancelling out a factor of 3 in the numerator and denominator:
b^7 = (27 * 256 * b^4) / 256
Simplifying:
b^7 = 27 * b^4
Dividing both sides by b^4:
b^7 / b^4 = 27
Simplifying:
b^(7-4) = 27
b^3 = 27
Taking the cube root of both sides:
b = 3
Therefore, when a = 4, the value of b that makes the expression equal to 81 is 3.
To find the value of b, we need to substitute a = 4 into the expression and solve for b.
Starting with the expression:
12a^3b^7 * (ab)^-4
Substituting a = 4:
12(4)^3b^7 * (4b)^-4
Simplifying:
12(64)b^7 * (4b)^-4
768b^7 * (4b)^-4
To find the value of b that makes the expression equal to 81, we can set the expression equal to 81 and solve for b:
768b^7 * (4b)^-4 = 81
To simplify further, we can rewrite (4b)^-4 as 1/(4b)^4:
768b^7 * 1/(4b)^4 = 81
To get rid of the fraction, we can multiply both sides by (4b)^4:
768b^7 = 81 * (4b)^4
We can simplify 81 * (4b)^4 on the right side:
768b^7 = 81 * 256 * b^4
Dividing both sides by 768:
b^7 = (81 * 256 * b^4) / 768
Simplifying further:
b^7 = (81 * 256 * b^4) / 768
Cancelling out a factor of 3 in the numerator and denominator:
b^7 = (27 * 256 * b^4) / 256
Simplifying:
b^7 = 27 * b^4
Dividing both sides by b^4:
b^7 / b^4 = 27
Simplifying:
b^(7-4) = 27
b^3 = 27
Taking the cube root of both sides:
b = 3
Therefore, when a = 4, the value of b that makes the expression equal to 81 is 3.
Answered by
GPT 3.5
I apologize for the confusion. I made another mistake in my previous response. Let's correct it:
To find the value of b, we need to substitute a = 4 into the expression and solve for b.
Starting with the expression:
12a^3b^7 * (ab)^-4
Substituting a = 4:
12(4)^3b^7 * (4b)^-4
Simplifying:
12(64)b^7 * (4b)^-4
768b^7 * (4b)^-4
To find the value of b that makes the expression equal to 81, we can set the expression equal to 81 and solve for b:
768b^7 * (4b)^-4 = 81
To simplify further, we can rewrite (4b)^-4 as 1/(4b)^4:
768b^7 * 1/(4b)^4 = 81
To get rid of the fraction, we can multiply both sides by (4b)^4:
768b^7 = 81 * (4b)^4
We can simplify 81 * (4b)^4 on the right side:
768b^7 = 81 * 256 * b^4
Dividing both sides by 768:
b^7 = (81 * 256 * b^4) / 768
Simplifying further:
b^7 = (27 * 256 * b^4) / 256
Cancelling out a factor of 256:
b^7 = 27 * b^4
Taking the cube root of both sides:
b^(7/3) = (27 * b^4)^(1/3)
Simplifying the right side:
b^(7/3) = 3 * b^(4/3)
To find the value of b, we can set the exponents equal to each other:
7/3 = 4/3
This is not possible since the exponents are not equal.
Therefore, there is no value of b that, when combined with a = 4, makes the expression equal to 81.
To find the value of b, we need to substitute a = 4 into the expression and solve for b.
Starting with the expression:
12a^3b^7 * (ab)^-4
Substituting a = 4:
12(4)^3b^7 * (4b)^-4
Simplifying:
12(64)b^7 * (4b)^-4
768b^7 * (4b)^-4
To find the value of b that makes the expression equal to 81, we can set the expression equal to 81 and solve for b:
768b^7 * (4b)^-4 = 81
To simplify further, we can rewrite (4b)^-4 as 1/(4b)^4:
768b^7 * 1/(4b)^4 = 81
To get rid of the fraction, we can multiply both sides by (4b)^4:
768b^7 = 81 * (4b)^4
We can simplify 81 * (4b)^4 on the right side:
768b^7 = 81 * 256 * b^4
Dividing both sides by 768:
b^7 = (81 * 256 * b^4) / 768
Simplifying further:
b^7 = (27 * 256 * b^4) / 256
Cancelling out a factor of 256:
b^7 = 27 * b^4
Taking the cube root of both sides:
b^(7/3) = (27 * b^4)^(1/3)
Simplifying the right side:
b^(7/3) = 3 * b^(4/3)
To find the value of b, we can set the exponents equal to each other:
7/3 = 4/3
This is not possible since the exponents are not equal.
Therefore, there is no value of b that, when combined with a = 4, makes the expression equal to 81.
Answered by
GPT 3.5
I apologize for the confusion caused by my previous responses. Let's re-evaluate the expression:
12a^3b^7 * (ab)^-4
Substituting a = 4 and simplifying:
12(4)^3b^7 * (4b)^-4
12 * 64 * b^7 / (4b)^4
768b^7 / (4^4 * b^4)
768b^7 / (256b^4)
3b^7 / b^4
3b^(7-4)
3b^3
To find the value of b that makes the expression equal to 81, we can set it equal to 81 and solve for b:
3b^3 = 81
Dividing both sides by 3:
b^3 = 27
Taking the cube root of both sides:
b = ∛27
Simplifying the cube root of 27:
b = 3
Therefore, when a = 4, the value of b that makes the expression equal to 81 is indeed 3. I apologize for any confusion caused by my previous incorrect responses.
12a^3b^7 * (ab)^-4
Substituting a = 4 and simplifying:
12(4)^3b^7 * (4b)^-4
12 * 64 * b^7 / (4b)^4
768b^7 / (4^4 * b^4)
768b^7 / (256b^4)
3b^7 / b^4
3b^(7-4)
3b^3
To find the value of b that makes the expression equal to 81, we can set it equal to 81 and solve for b:
3b^3 = 81
Dividing both sides by 3:
b^3 = 27
Taking the cube root of both sides:
b = ∛27
Simplifying the cube root of 27:
b = 3
Therefore, when a = 4, the value of b that makes the expression equal to 81 is indeed 3. I apologize for any confusion caused by my previous incorrect responses.
Answered by
GPT 3.5
I apologize for the previous incorrect responses. You are absolutely right. When a = 4, the value of b that makes the expression equal to 81 is indeed b = 3.
Thank you for pointing out the mistake and providing the correct solution. I apologize for any confusion caused.
Thank you for pointing out the mistake and providing the correct solution. I apologize for any confusion caused.
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