Question

calculate the value of (left(7^{\frac{4}{3}}
ight)^{\frac{5}{4}})

Explanation:

Step1: Simplify the base

First, simplify the fraction in the base. The base is \(7^{\frac{4}{3}}\), and we are raising it to the power of \(\frac{5}{4}\). When raising a power to a power, we multiply the exponents: \((a^m)^n=a^{m\times n}\). So we multiply \(\frac{4}{3}\) and \(\frac{5}{4}\).
\(\frac{4}{3}\times\frac{5}{4}=\frac{4\times5}{3\times4}=\frac{5}{3}\) (the 4s cancel out)

Step2: Rewrite the expression

Now the expression becomes \(7^{\frac{5}{3}}\). We can also write this as the cube - root of \(7^5\) or as \(7\times7^{\frac{2}{3}}\), but if we want to write it in the form of a single exponent or a radical, \(7^{\frac{5}{3}}=\sqrt[3]{7^5}=\sqrt[3]{16807}\) (since \(7^5 = 7\times7\times7\times7\times7=16807\)) or we can keep it in exponential form \(7^{\frac{5}{3}}\). If we calculate the numerical value: \(7^{\frac{5}{3}}=(7^{\frac{1}{3}})^5\). \(7^{\frac{1}{3}}\approx1.913\), and \((1.913)^5\approx1.913\times1.913 = 3.66, 3.66\times1.913\approx7.00, 7.00\times1.913\approx13.39, 13.39\times1.913\approx25.62\) (approximate value). But if we just simplify the exponent:
After multiplying the exponents \(\frac{4}{3}\times\frac{5}{4}=\frac{5}{3}\), so the expression simplifies to \(7^{\frac{5}{3}}\) or \(\sqrt[3]{7^{5}}\) or \(\sqrt[3]{16807}\) or approximately \(25.62\) (if we calculate the decimal).

Answer:

\(7^{\frac{5}{3}}\) (or \(\sqrt[3]{7^{5}}\), \(\sqrt[3]{16807}\), or the approximate decimal value \(25.62\))