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For the following exercise, solve the rational exponent equation. Use factoring where necessary.

[tex] x^{\frac{5}{4}} = 3125 [/tex]

Solve for [tex] x [/tex]:

[tex] x = [/tex]

Answer :

To solve the equation [tex]\( x^{\frac{5}{4}} = 3125 \)[/tex], we need to find the value of [tex]\( x \)[/tex].

Here's a step-by-step explanation:

1. Start with the equation given:
[tex]\[
x^{\frac{5}{4}} = 3125
\][/tex]

2. To isolate [tex]\( x \)[/tex], we need to get rid of the exponent [tex]\(\frac{5}{4}\)[/tex]. We can do this by raising both sides of the equation to the reciprocal of [tex]\(\frac{5}{4}\)[/tex], which is [tex]\(\frac{4}{5}\)[/tex]. So, raise both sides to the power of [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
(x^{\frac{5}{4}})^{\frac{4}{5}} = 3125^{\frac{4}{5}}
\][/tex]

3. When you raise a power to another power, you multiply the exponents. Thus, the left side becomes:
[tex]\[
x^{1} = x
\][/tex]

4. Now the equation simplifies to:
[tex]\[
x = 3125^{\frac{4}{5}}
\][/tex]

5. Calculate [tex]\( 3125^{\frac{4}{5}} \)[/tex]. After evaluating, we find:
[tex]\[
x = 625
\][/tex]

Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 625 \)[/tex].

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