Question

question
express \\(\left(10^{\frac{1}{4}}\
ight)^3\\) in simplest radical form.

Explanation:

Step1: Apply exponent power rule

When raising a power to a power, we multiply the exponents. The formula is \((a^{m})^{n}=a^{m\times n}\). Here, \(a = 10\), \(m=\frac{1}{4}\), and \(n = 3\). So we calculate \(\frac{1}{4}\times3=\frac{3}{4}\), and the expression becomes \(10^{\frac{3}{4}}\).

Step2: Convert exponent to radical form

Recall that \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (where \(n\gt0\) and \(a\geq0\) for real - valued roots). For \(a = 10\), \(m = 3\), and \(n = 4\), we have \(10^{\frac{3}{4}}=\sqrt[4]{10^{3}}\).

Step3: Simplify the radicand

Calculate \(10^{3}=10\times10\times10 = 1000\). So \(\sqrt[4]{10^{3}}=\sqrt[4]{1000}\).

Answer:

\(\sqrt[4]{1000}\) (or we can also write it as \(10^{\frac{3}{4}}\), but the radical form \(\sqrt[4]{1000}\) is more in line with the "simplest radical form" requirement)