Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 128

1장, 문제 128

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.
$7 (5 t + 3)^{- \frac{5}{3}} + 14 (5 t + 3)^{- \frac{2}{3}} - 21 (5 t + 3)^{\frac{1}{3}}$

검증된 단계별 안내

Identify the common variable expression in all terms, which is

(^{5 t + 3) \frac{-5}{3}}

(^{5 t + 3) \frac{-2}{3}}

, and

(^{5 t + 3) \frac{1}{3}}

Determine the least power (smallest exponent) among the powers of

(5 t + 3)

. The exponents are

\frac{-5}{3}

\frac{-2}{3}

, and

\frac{1}{3}

. The least power is

\frac{-5}{3}

Factor out

(^{5 t + 3) \frac{-5}{3}}

from each term by rewriting each term as a product of this factor and another power of

(5 t + 3)

For each term, subtract the exponent of the factored out term from the original exponent to find the new exponent inside the parentheses after factoring. For example, for the second term, subtract

\frac{-5}{3}

from

\frac{-2}{3}

to get the new exponent.

Rewrite the expression as the product of the factored out term and the sum of the remaining terms with their adjusted exponents, keeping the coefficients intact.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Factoring Expressions with Variable Exponents

Factoring expressions with variable exponents involves identifying the smallest exponent among terms and factoring it out. This simplifies the expression by reducing the powers inside the parentheses, making it easier to combine or simplify further.

Properties of Exponents

Understanding exponent rules, such as how to add, subtract, and factor exponents, is essential. For example, factoring out a term with an exponent means subtracting that exponent from each term's exponent inside the expression.

Assumption of Positive Variables

Assuming variables represent positive real numbers allows the use of exponent rules without worrying about undefined expressions or complex values. This assumption ensures that expressions with fractional or negative exponents are valid and simplifies the factoring process.

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8.7(5t+3)−53+14(5t+3)−23−21(5t+3)137(5t+3)^{-\(\frac\)53}+14(5t+3)^{-\(\frac\)23}-21(5t+3)^{\(\frac\)13}