Question

1. rewrite in a different form (reescribe de otra forma):

a) ( x^{\frac{3}{4}} )
b) ( sqrt4{a^5} )

Explanation:

Step1: Recall exponent - root relation

The formula that relates exponents and roots is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (or equivalently \((\sqrt[n]{a})^{m}\)).

Step2: Rewrite \(x^{\frac{3}{4}}\)

For the expression \(x^{\frac{3}{4}}\), using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), where \(a = x\), \(m = 3\) and \(n=4\), we can rewrite it as \(\sqrt[4]{x^{3}}\) (or \((\sqrt[4]{x})^{3}\)).

Step3: Rewrite \(\sqrt[4]{a^{5}}\)

First, we can express the exponent of \(a\) in the radicand as a sum of an integer and a fraction. We know that \(a^{5}=a^{4 + 1}=a^{4}\times a^{1}\). Then \(\sqrt[4]{a^{5}}=\sqrt[4]{a^{4}\times a}=\sqrt[4]{a^{4}}\times\sqrt[4]{a}\). Since \(\sqrt[4]{a^{4}}=a\) (for \(a\geq0\)), we have \(\sqrt[4]{a^{5}}=a\sqrt[4]{a}\). Also, using the exponent - root formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), for \(\sqrt[4]{a^{5}}\), we have \(m = 5\) and \(n = 4\), so it can be written as \(a^{\frac{5}{4}}\) (because \(a^{\frac{5}{4}}=\sqrt[4]{a^{5}}\)).

Answer:

• For \(x^{\frac{3}{4}}\): \(\sqrt[4]{x^{3}}\) (or \((\sqrt[4]{x})^{3}\))
• For \(\sqrt[4]{a^{5}}\): \(a^{\frac{5}{4}}\) (or \(a\sqrt[4]{a}\))