Question
Solve the equation with rational exponents. Check all proposed solutions.
x Superscript five halves Baseline equals 32
x Superscript five halves Baseline equals 32
Answers
GPT-4o mini
To solve the equation \( x^{\frac{5}{2}} = 32 \), we can first isolate \( x \) by raising both sides of the equation to the reciprocal of the exponent, which is \( \frac{2}{5} \).
1. Raise both sides to the power of \( \frac{2}{5} \):
\[
\left( x^{\frac{5}{2}} \right)^{\frac{2}{5}} = 32^{\frac{2}{5}}
\]
2. Simplify the left side:
\[
x^{\left( \frac{5}{2} \cdot \frac{2}{5} \right)} = x^1 = x
\]
3. Now calculate the right side \( 32^{\frac{2}{5}} \):
First, express 32 as a power of 2:
\[
32 = 2^5
\]
Then,
\[
32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^{5 \cdot \frac{2}{5}} = 2^2 = 4
\]
4. Therefore, we have:
\[
x = 4
\]
Now, we will check our solution by substituting \( x = 4 \) back into the original equation:
\[
(4)^{\frac{5}{2}} = 32
\]
Calculating \( (4)^{\frac{5}{2}} \):
\[
(4)^{\frac{5}{2}} = (2^2)^{\frac{5}{2}} = 2^{2 \cdot \frac{5}{2}} = 2^5 = 32
\]
Since both sides are equal, our solution is confirmed.
Therefore, the solution to the equation \( x^{\frac{5}{2}} = 32 \) is
\[
\boxed{4}
\]
1. Raise both sides to the power of \( \frac{2}{5} \):
\[
\left( x^{\frac{5}{2}} \right)^{\frac{2}{5}} = 32^{\frac{2}{5}}
\]
2. Simplify the left side:
\[
x^{\left( \frac{5}{2} \cdot \frac{2}{5} \right)} = x^1 = x
\]
3. Now calculate the right side \( 32^{\frac{2}{5}} \):
First, express 32 as a power of 2:
\[
32 = 2^5
\]
Then,
\[
32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^{5 \cdot \frac{2}{5}} = 2^2 = 4
\]
4. Therefore, we have:
\[
x = 4
\]
Now, we will check our solution by substituting \( x = 4 \) back into the original equation:
\[
(4)^{\frac{5}{2}} = 32
\]
Calculating \( (4)^{\frac{5}{2}} \):
\[
(4)^{\frac{5}{2}} = (2^2)^{\frac{5}{2}} = 2^{2 \cdot \frac{5}{2}} = 2^5 = 32
\]
Since both sides are equal, our solution is confirmed.
Therefore, the solution to the equation \( x^{\frac{5}{2}} = 32 \) is
\[
\boxed{4}
\]