How do you simplify 4 to the power of 2-3-

Step 1: Use the Fractional Exponent rule:Given that, 4^2/3The fractional Exponent rule is a^m/n=√(a^m). Where a is the base integer, m is the numerator and n is ...

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Standard X

Mathematics

Rational Expression

How do you si...

Question

How do you simplify $4$ to the power of $\frac{2}{3}$ ?

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Solution

Step-1: Use the Fractional Exponent rule:
Given that, $4^{\frac{2}{3}}$
The fractional Exponent rule is $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ . Where $a$ is the base integer, $m$ is the numerator and $n$ is the denominator.
Substitute $a = 4, m = 2 and n = 3$ in the above rule:
$\begin{array}{rcl} \Rightarrow & a^{\frac{m}{n}} = \sqrt[n]{a^{m}} \\ \Rightarrow & 4^{\frac{2}{3}} = \sqrt[3]{4^{2}} \\ = & \sqrt[3]{16} \\ = & \sqrt[3]{8 (2)} [Factor 8 out of 16] \\ ∵ & = & \sqrt[3]{2^{3} \times 2} [Rewrite 8 as 2^{3}] \end{array}$
Step-2: Apply the radical rule:
Use the radical rule $\sqrt[n]{a b} = \sqrt[n]{a} \sqrt[n]{b}$ :
$\begin{array}{rcl} \Rightarrow & \sqrt[3]{2^{3} \times 2} = \sqrt[3]{2^{3}} \sqrt[3]{2} \\ = & 2 \sqrt[3]{2} [∴ \sqrt[n]{a^{n}} = a] \end{array}$
Hence, the simplification of $4^{\frac{2}{3}}$ is $2 \sqrt[3]{2}$ .

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