To rewrite the expression [tex]\( 2 \sqrt[4]{x^7} \)[/tex] using rational exponents, follow these steps:
1. Recall that the fourth root of a number is equivalent to raising that number to the power of [tex]\(\frac{1}{4}\)[/tex]. In other words, [tex]\(\sqrt[4]{y} = y^{\frac{1}{4}}\)[/tex].
2. Apply this property to the expression inside the fourth root. Here, [tex]\( \sqrt[4]{x^7} \)[/tex] means [tex]\( (x^7)^{\frac{1}{4}} \)[/tex].
3. Use the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to combine the exponents. For [tex]\( (x^7)^{\frac{1}{4}} \)[/tex], you multiply the exponents [tex]\( 7 \)[/tex] and [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[
(x^7)^{\frac{1}{4}} = x^{7 \cdot \frac{1}{4}} = x^{\frac{7}{4}}
\][/tex]
4. Now, rewrite the original expression [tex]\( 2 \sqrt[4]{x^7} \)[/tex] using the result from step 3:
[tex]\[
2 \sqrt[4]{x^7} = 2 \cdot x^{\frac{7}{4}}
\][/tex]
Therefore, the expression [tex]\( 2 \sqrt[4]{x^7} \)[/tex] rewritten using rational exponents is:
[tex]\[
2x^{\frac{7}{4}}
\][/tex]
