Question

score: 3/10 penalty: none question simplify \\(\sqrt4{81x^{32}}\\) completely given \\(x > 0\\). answer attempt 1 out of 2

Explanation:

Step1: Simplify the constant term

We know that \( 81 = 3^4 \), so we can rewrite the fourth - root of 81 as \( \sqrt[4]{3^4} \). By the property of \( n \) - th roots, \( \sqrt[n]{a^n}=a \) when \( n \) is even and \( a\geq0 \). Here, \( n = 4 \) and \( a = 3 \), so \( \sqrt[4]{3^4}=3 \).

Step2: Simplify the variable term

For the term with \( x \), we have \( x^{32} \). We want to find the fourth - root of \( x^{32} \), that is \( \sqrt[4]{x^{32}} \). Using 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 of radicals \( \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b} \) (where \( a = 81 \) and \( b=x^{32} \) and \( n = 4 \)), we have \( \sqrt[4]{81x^{32}}=\sqrt[4]{81}\cdot\sqrt[4]{x^{32}} \). Substituting the values we found in Step 1 and Step 2, we get \( 3\cdot x^{8}=3x^{8} \).

Answer:

\( 3x^{8} \)