To write the expression [tex]\(\sqrt[3]{a^2 + b^2}\)[/tex] without radicals and using only positive exponents, we can follow these steps:
1. Understand the expression in terms of radicals:
The given expression is [tex]\(\sqrt[3]{a^2 + b^2}\)[/tex]. This is a cube root expression.
2. Convert the radical to a rational exponent:
Recall that a cube root can be written as an exponent of [tex]\(\frac{1}{3}\)[/tex]. Specifically, for any expression [tex]\(x\)[/tex], [tex]\(\sqrt[3]{x}\)[/tex] is equivalent to [tex]\(x^{\frac{1}{3}}\)[/tex].
3. Apply this rule to the given expression:
For the given expression [tex]\(\sqrt[3]{a^2 + b^2}\)[/tex]:
[tex]\[
\sqrt[3]{a^2 + b^2} = (a^2 + b^2)^{\frac{1}{3}}
\][/tex]
Therefore, the expression [tex]\(\sqrt[3]{a^2 + b^2}\)[/tex] written without radicals and using only positive exponents is:
[tex]\[
(a^2 + b^2)^{0.333333333333333}
\][/tex]
