Find each root.81p12q44\sqrt[4]{$$81p^{12}$q^4$$} 481p12q4

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 42

1장, 문제 42

Find each root. $$\sqrt$[4]{81p^{12}q^4}$

검증된 단계별 안내

Recognize that the expression involves a fourth root, which can be written as an exponent of $ \frac{1}{4} $. So, rewrite the expression $ \sqrt[4]{81p^{12}q^{4}} $ as $ (81p^{12}q^{4})^{\frac{1}{4}} $.

Apply the exponent $ \frac{1}{4} $ to each factor inside the parentheses separately: $ 81^{\frac{1}{4}} $, $ (p^{12})^{\frac{1}{4}} $, and $ (q^{4})^{\frac{1}{4}} $.

Simplify each term using the power of a power rule $ (a^{m})^{n} = a^{mn} $: $ 81^{\frac{1}{4}} $, $ p^{12 \times \frac{1}{4}} = p^{3} $, and $ q^{4 \times \frac{1}{4}} = q^{1} = q $.

Evaluate $ 81^{\frac{1}{4}} $ by recognizing that 81 is a perfect power of 3: $ 81 = 3^{4} $, so $ 81^{\frac{1}{4}} = 3 $.

Combine all simplified parts to write the final expression for the root as $ 3p^{3}q $.

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주요 개념

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

Nth Root of a Number

The nth root of a number is a value that, when raised to the power n, equals the original number. For example, the fourth root (∜) of 81 is the number that raised to the 4th power gives 81. Understanding how to find roots is essential for simplifying expressions involving radicals.

Properties of Exponents

Exponents indicate repeated multiplication, and their properties help simplify expressions. When taking roots, exponents can be divided by the root's index (e.g., the fourth root of p¹² is p^(12/4) = p³). This rule applies to variables and constants alike.

Simplifying Radical Expressions

Simplifying radicals involves rewriting expressions to their simplest form by factoring and applying root and exponent rules. For example, breaking down 81 into 3⁴ helps find its fourth root easily. This process makes expressions easier to interpret and use.

Find each root.81p12q44\(\sqrt\)[4]{81p^{12}q^4} 481p12q4

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Nth Root of a Number

Properties of Exponents

Simplifying Radical Expressions

Find each root. $\(\sqrt\)[4]{81p^{12}q^4}$