Question

question
simplify (sqrt3{125x^{24}}) completely given (x > 0).
answer attempt 1 out of 2

Explanation:

Step1: Simplify the cube root of 125

We know that \( 5^3 = 125 \), so \( \sqrt[3]{125}=\sqrt[3]{5^3} = 5 \).

Step2: Simplify the cube root of \( x^{24} \)

Using the property of exponents \( \sqrt[n]{a^m}=a^{\frac{m}{n}} \), for \( \sqrt[3]{x^{24}} \), we have \( x^{\frac{24}{3}}=x^8 \) (since \( x>0 \), we don't need to consider absolute value here).

Step3: Combine the results

By the property of radicals \( \sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b} \) (where \( a = 125 \) and \( b=x^{24} \)), we get \( \sqrt[3]{125x^{24}}=\sqrt[3]{125}\cdot\sqrt[3]{x^{24}} = 5\cdot x^8=5x^8 \).

Answer:

\( 5x^8 \)