Question

question
simplify (sqrt4{256x^{32}}) completely given (x > 0).

answer attempt 1 out of 5

Explanation:

Step1: Simplify the constant term

Recall that \( \sqrt[4]{a^4}=a \) for \( a\geq0 \). We know that \( 256 = 4^4 \), so \( \sqrt[4]{256}=\sqrt[4]{4^4} = 4 \).

Step2: Simplify the variable term

For the variable part \( \sqrt[4]{x^{32}} \), use the property of exponents \( \sqrt[n]{a^m}=a^{\frac{m}{n}} \). Here \( n = 4 \) and \( m=32 \), so \( \sqrt[4]{x^{32}}=x^{\frac{32}{4}}=x^{8} \).

Step3: Combine the results

Using the property \( \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b} \) (for \( a\geq0,b\geq0 \)), we have \( \sqrt[4]{256x^{32}}=\sqrt[4]{256}\cdot\sqrt[4]{x^{32}} \). Substituting the values from Step 1 and Step 2, we get \( 4\cdot x^{8}=4x^{8} \).

Answer:

\( 4x^{8} \)