Question
Solve the following equation with a rational exponent: x^2/3=64
.(1 point)
Responses
32
32
16
16
512
512
8
8
.(1 point)
Responses
32
32
16
16
512
512
8
8
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To solve the equation \( x^{2/3} = 64 \), we can first eliminate the rational exponent by raising both sides of the equation to the reciprocal of the exponent \( \frac{3}{2} \):
\[
(x^{2/3})^{3/2} = 64^{3/2}
\]
This simplifies to:
\[
x^{(2/3) \cdot (3/2)} = x^1 = x
\]
Now we need to calculate \(64^{3/2}\). We can do this in two steps:
1. First, find \(64^{1/2}\):
\[
64^{1/2} = \sqrt{64} = 8
\]
2. Then raise the result to the power of 3:
\[
64^{3/2} = (64^{1/2})^3 = 8^3 = 512
\]
Therefore:
\[
x = 512
\]
So the solution is:
\[
\boxed{512}
\]
\[
(x^{2/3})^{3/2} = 64^{3/2}
\]
This simplifies to:
\[
x^{(2/3) \cdot (3/2)} = x^1 = x
\]
Now we need to calculate \(64^{3/2}\). We can do this in two steps:
1. First, find \(64^{1/2}\):
\[
64^{1/2} = \sqrt{64} = 8
\]
2. Then raise the result to the power of 3:
\[
64^{3/2} = (64^{1/2})^3 = 8^3 = 512
\]
Therefore:
\[
x = 512
\]
So the solution is:
\[
\boxed{512}
\]
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