Find each root.x66\sqrt[6]{$$x^6$$} 6x6

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 40

1장, 문제 40

Find each root. $$\sqrt$[6]{x^6}$

검증된 단계별 안내

Recognize that the expression ⁶√x^6 represents the sixth root of x raised to the sixth power, which can be written as $\sqrt[6]{x^6}$.

Recall the property of radicals and exponents: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Applying this, rewrite the expression as $x^{\frac{6}{6}}$.

Simplify the exponent $\frac{6}{6}$ to 1, so the expression becomes $x^1$, which is simply $x$.

Consider the domain of the original expression. Since the sixth root is an even root, the radicand $x^6$ is always non-negative for all real $x$, so the expression is defined for all real numbers.

Therefore, the root of the expression $\sqrt[6]{x^6}$ simplifies to $|x|$, the absolute value of $x$, because even roots return the principal (non-negative) root.

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54s

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주요 개념

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

Nth Roots and Radicals

The nth root of a number is a value that, when raised to the power n, gives the original number. For example, the 6th root of x^6 means finding a number which, when raised to the 6th power, equals x^6. Understanding how to simplify and interpret nth roots is essential for solving such expressions.

Properties of Exponents

Exponents represent repeated multiplication, and their properties allow simplification of expressions. Specifically, (x^a)^(b) = x^(a*b). This property helps simplify the expression ⁶√x^6 by rewriting the root as a fractional exponent, enabling easier calculation of the root.

Absolute Value in Even Roots

When taking even roots (like square roots or 6th roots) of even powers, the result is the absolute value of the base because even powers eliminate sign information. For example, ⁶√x^6 equals |x|, ensuring the root is non-negative regardless of x's sign.

Find each root.x66\(\sqrt\)[6]{x^6} 6x6

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Nth Roots and Radicals

Properties of Exponents

Absolute Value in Even Roots

Find each root. $\(\sqrt\)[6]{x^6}$