Question

1. \\(\left(14^{3}\

ight)^{\frac{1}{2}}\\)

Explanation:

Step1: Apply exponent power rule

The power of a power rule states that \((a^m)^n = a^{m\times n}\). Here, \(a = 14\), \(m = 3\), and \(n=\frac{1}{2}\). So we multiply the exponents: \(3\times\frac{1}{2}=\frac{3}{2}\).

Step2: Rewrite the exponent

We can rewrite \(14^{\frac{3}{2}}\) as \(14^{1+\frac{1}{2}}\). Using the property \(a^{m + n}=a^m\times a^n\), we get \(14^1\times14^{\frac{1}{2}}\). Since \(14^{\frac{1}{2}}=\sqrt{14}\), this becomes \(14\sqrt{14}\). Alternatively, we can keep it in exponential form as \(14^{\frac{3}{2}}\) or calculate the numerical value: \(14^{\frac{3}{2}}=\sqrt{14^3}=\sqrt{2744}\approx52.38\). But the simplified exponential form or the radical form is more precise.

Answer:

\(14^{\frac{3}{2}}\) (or \(14\sqrt{14}\) or approximately \(52.38\))