Questions: Evaluate the expression without using a calculator. 125^(2/3) Rewrite the expression using a radical. 125^(2/3)=( Do not evaluate.)

Transcript text: Evaluate the expression without using a calculator. \[ 125^{\frac{2}{3}} \] Rewrite the expression using a radical. \[ 125^{\frac{2}{3}}=\square(\text { Do not evaluate.) } \]

Solution

Solution Steps

Step 1: Rewrite the exponent as a radical

The expression \(125^{\frac{2}{3}}\) can be rewritten using radicals. Recall that \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). Applying this rule: \[ 125^{\frac{2}{3}} = \sqrt[3]{125^2} \]

Step 2: Simplify the expression inside the radical

First, calculate \(125^2\): \[ 125^2 = 125 \times 125 = 15625 \] So, the expression becomes: \[ \sqrt[3]{15625} \]

Step 3: Evaluate the cube root

Now, evaluate \(\sqrt[3]{15625}\). Since \(25^3 = 15625\), the cube root of 15625 is 25. Thus: \[ \sqrt[3]{15625} = 25 \]

Step 4: Rewrite the expression using a radical (without evaluating)

The expression \(125^{\frac{2}{3}}\) can be rewritten as: \[ 125^{\frac{2}{3}} = \sqrt[3]{125^2} \]