Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√x² • ⁵√x³

검증된 단계별 안내

Identify the operation: You are multiplying two fifth roots, \(\sqrt[5]{x^2}\) and \(\sqrt[5]{x^3}\).

Recall the property of radicals that allows multiplication under the same root: \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\).

Apply this property to combine the radicals: \(\sqrt[5]{x^2} \cdot \sqrt[5]{x^3} = \sqrt[5]{x^2 \cdot x^3}\).

Use the laws of exponents to multiply inside the radical: \(x^2 \cdot x^3 = x^{2+3} = x^5\).

Rewrite the expression as \(\sqrt[5]{x^5}\) and recognize that the fifth root of \(x^5\) simplifies to \(x\) because \(\sqrt[5]{x^5} = x^{5/5} = x\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Radicals

Radicals represent roots of numbers or expressions. The nth root of a product equals the product of the nth roots, allowing multiplication under a single radical. For example, ⁿ√a × ⁿ√b = ⁿ√(a × b), which simplifies operations involving radicals with the same index.

Multiplication of Radicals with the Same Index

When multiplying radicals that have the same root index, you can combine them into one radical by multiplying their radicands. For instance, ⁵√x² × ⁵√x³ = ⁵√(x² × x³) = ⁵√x⁵, simplifying the expression efficiently.

Simplifying Radicals Using Exponent Rules

Radicals can be expressed as fractional exponents, where ⁿ√x^m = x^(m/n). This allows simplification by applying exponent rules, such as adding exponents when multiplying like bases, making it easier to simplify expressions like ⁵√x⁵ = x^(5/5) = x.