Questions: Simplify completely. 625^(3 / 4) Which term represents the simplified form of 625^(3 / 4) ? A. (3√(√(2) 5^4)) B. ((4√625)^3) C. 125 D. (4√625^3)

Transcript text: Simplify completely. $625^{3 / 4}$ Which term represents the simplified form of $625^{3 / 4}$ ? A. $\sqrt[3]{\sqrt{2} 5^{4}}$ B. $(\sqrt[4]{625})^{3}$ C. 125 D. $\sqrt[4]{625^{3}}$

Solution

Solution Steps

Step 1: Rewrite the exponent in radical form

The expression $625^{3/4}$ can be rewritten using the property of exponents $a^{m/n} = \sqrt[n]{a^m}$. Applying this property: \[ 625^{3/4} = \sqrt[4]{625^3} \]

Step 2: Simplify the radical expression

First, simplify $625^3$. Since $625 = 5^4$, we have: \[ 625^3 = (5^4)^3 = 5^{12} \] Now, substitute back into the radical: \[ \sqrt[4]{625^3} = \sqrt[4]{5^{12}} \]

Step 3: Simplify the fourth root

Using the property $\sqrt[n]{a^m} = a^{m/n}$, simplify the fourth root: \[ \sqrt[4]{5^{12}} = 5^{12/4} = 5^3 \] Finally, calculate $5^3$: \[ 5^3 = 125 \]

Step 4: Match the simplified form with the given options

The simplified form of $625^{3/4}$ is $125$, which corresponds to option C.