Simplify each radical. Assume all variables represent positive real numbers. $$\sqrt$[6]{x^{18}y^2}$

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Identify the expression inside the sixth root: $\sqrt[6]{x^{18} y^{2}}$.

Recall the property of radicals that $\sqrt[n]{a^{m}} = a^{\frac{m}{n}}$. Apply this to each variable inside the radical: $\sqrt[6]{x^{18}} = x^{\frac{18}{6}}$ and $\sqrt[6]{y^{2}} = y^{\frac{2}{6}}$.

Simplify the exponents by dividing: $x^{\frac{18}{6}} = x^{3}$ and $y^{\frac{2}{6}} = y^{\frac{1}{3}}$.

Rewrite the expression as a product of powers: $x^{3} y^{\frac{1}{3}}$.

Express the fractional exponent back as a radical if desired: $y^{\frac{1}{3}} = \sqrt[3]{y}$. So the simplified form is $x^{3} \sqrt[3]{y}$.

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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 us to separate and simplify each factor under the radical individually.

Exponents and Roots Relationship

An nth root can be expressed as a fractional exponent 1/n. For example, the sixth root of x¹⁸ is x raised to the power 18/6, which simplifies the expression by converting roots into exponents.

Simplifying Expressions with Variables

When simplifying radicals with variables, divide the exponent by the root index to find the simplified power. Since variables are positive, no absolute value is needed, and leftover exponents remain under the radical.

Simplify each radical. Assume all variables represent positive real numbers. x18y26\(\sqrt\)[6]{x^{18}y^2} 6x18y2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Properties of Radicals

Exponents and Roots Relationship

Simplifying Expressions with Variables

Simplify each radical. Assume all variables represent positive real numbers. $\(\sqrt\)[6]{x^{18}y^2}$