Simplify the expression. $$ \left(z^{\frac{5}{4}} \cdot y^{-2}\right)^{\frac{1}{5}} $$ Write your answer without using negative exponents. Assume that all variables are positive real numbers.

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UpStudy AI · Accepted Answer

We start with the expression

$$
\left(z^{\frac{5}{4}} \cdot y^{-2}\right)^{\frac{1}{5}}.
$$

**Step 1:** Apply the exponent rule $$\left(ab\right)^c = a^c \cdot b^c$$:

$$
\left(z^{\frac{5}{4}}\right)^{\frac{1}{5}} \cdot \left(y^{-2}\right)^{\frac{1}{5}}.
$$

**Step 2:** Use the power rule $$\left(a^b\right)^c = a^{b \cdot c}$$ on both terms:

$$
z^{\frac{5}{4} \cdot \frac{1}{5}} \cdot y^{-2 \cdot \frac{1}{5}}.
$$

**Step 3:** Simplify the exponents:

$$
z^{\frac{5}{20}} \cdot y^{-\frac{2}{5}} = z^{\frac{1}{4}} \cdot y^{-\frac{2}{5}}.
$$

**Step 4:** Rewrite the expression without negative exponents, using the rule $$a^{-b} = \frac{1}{a^b}$$:

$$
\frac{z^{\frac{1}{4}}}{y^{\frac{2}{5}}}.
$$

Thus, the simplified expression is

$$
\frac{z^{\frac{1}{4}}}{y^{\frac{2}{5}}}.
$$
The simplified expression is $$ \frac{z^{\frac{1}{4}}}{y^{\frac{2}{5}}} $$.

Simplify the expression.

Write your answer without using negative exponents.
Assume that all variables are positive real numbers.

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Simplify the expression. Write your answer without using negative exponents. Assume that all variables are positive real numbers.