Question $\left(\frac{1}{27}\right)^{\frac{2}{3}}=$

Studdy Solution

STEP 1

1. We are asked to simplify the expression $\left(\frac{1}{27}\right)^{\frac{2}{3}}$ .
2. The expression involves a fractional exponent, which can be simplified using properties of exponents and roots.

STEP 2

1. Rewrite the expression using the property of exponents.
2. Simplify the expression by evaluating the root and the power.

STEP 3

Rewrite the expression $\left(\frac{1}{27}\right)^{\frac{2}{3}}$ using the property of exponents that states $a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^m$ .
$\left(\frac{1}{27}\right)^{\frac{2}{3}} = \left(\left(\frac{1}{27}\right)^{\frac{1}{3}}\right)^2$

STEP 4

Simplify $\left(\frac{1}{27}\right)^{\frac{1}{3}}$ by finding the cube root of $\frac{1}{27}$ .
$\left(\frac{1}{27}\right)^{\frac{1}{3}} = \frac{1}{\sqrt[3]{27}} = \frac{1}{3}$

STEP 5

Now, substitute back into the expression:
$\left(\frac{1}{3}\right)^2 = \frac{1}{3^2} = \frac{1}{9}$
The simplified expression is:
$\boxed{\frac{1}{9}}$

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