To transform the expression [tex]\(\left(\sqrt[3]{x^4}\right)^5\)[/tex] into an expression with a rational exponent, follow these steps:
1. Understand the cube root notation: The cube root of [tex]\(x^4\)[/tex] can be expressed as [tex]\(x^{4/3}\)[/tex]. This is because taking the cube root of a number is equivalent to raising that number to the power of [tex]\(1/3\)[/tex]. So, [tex]\(\sqrt[3]{x^4}\)[/tex] is equal to [tex]\(x^{4/3}\)[/tex].
2. Apply the exponent property: The given expression [tex]\(\left(\sqrt[3]{x^4}\right)^5\)[/tex] can now be rewritten with the rational exponent we found in the previous step. This gives us [tex]\(\left(x^{4/3}\right)^5\)[/tex].
3. Simplify using exponent multiplication: According to the laws of exponents, specifically the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we multiply the exponents. Here, we multiply [tex]\(4/3\)[/tex] by [tex]\(5\)[/tex] to get the new exponent. Performing the multiplication: [tex]\(\frac{4}{3} \times 5 = \frac{20}{3}\)[/tex].
4. Rewrite the expression with the simplified exponent: Now, we replace the original expression with the simplified exponent obtained in the previous step. Thus, [tex]\(\left(x^{4/3}\right)^5\)[/tex] simplifies to [tex]\(x^{20/3}\)[/tex].
Therefore, the transformed expression is [tex]\(x^{20/3}\)[/tex].
