Question

1. explain the meaning of $2^{\frac{3}{4}}$ in terms of the product rule.

Explanation:

Step1: Recall the product rule for exponents

The product rule states that \(a^{m + n}=a^{m}\times a^{n}\). Also, a fractional exponent \(a^{\frac{b}{c}}=\sqrt[c]{a^{b}}\). For \(2^{\frac{3}{4}}\), we can rewrite \(\frac{3}{4}\) as \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\).

Step2: Apply the product rule

By the product rule \(a^{m + n}=a^{m}\times a^{n}\), if \(a = 2\), \(m=\frac{1}{4}\), \(n=\frac{1}{4}\), and then adding another \(\frac{1}{4}\). So \(2^{\frac{3}{4}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^{\frac{1}{4}}\times2^{\frac{1}{4}}\times2^{\frac{1}{4}}\). And \(2^{\frac{1}{4}}=\sqrt[4]{2}\), so \(2^{\frac{3}{4}}=\sqrt[4]{2}\times\sqrt[4]{2}\times\sqrt[4]{2}\)

Answer:

\(2^{\frac{3}{4}}\) can be written as \(2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}\), and by the product rule \(a^{m + n}=a^{m}\times a^{n}\) (\(a = 2\), \(m=n=\frac{1}{4}\) in successive applications), \(2^{\frac{3}{4}}=2^{\frac{1}{4}}\times2^{\frac{1}{4}}\times2^{\frac{1}{4}}\), where \(2^{\frac{1}{4}}=\sqrt[4]{2}\)