Questions: Simplify the expression. Assume that a denotes any real number. sqrt[5]32 a^7

Solution Steps

The expression is \(\sqrt[5]{32 a^{7}}\). First, simplify the constant \(32\). Since \(32 = 2^5\), we can rewrite the expression as: \[ \sqrt[5]{2^5 a^{7}} \]

The fifth root of \(2^5\) is \(2\), because \(\sqrt[5]{2^5} = 2\). Thus, the expression becomes: \[ 2 \cdot \sqrt[5]{a^{7}} \]

Now, simplify \(\sqrt[5]{a^{7}}\). Since \(a^{7} = a^{5} \cdot a^{2}\), we can rewrite the expression as: \[ 2 \cdot \sqrt[5]{a^{5} \cdot a^{2}} \] The fifth root of \(a^{5}\) is \(a\), because \(\sqrt[5]{a^{5}} = a\). Therefore, the expression simplifies to: \[ 2a \cdot \sqrt[5]{a^{2}} \]

Final Answer