Simplify each exponential expression in Exercises 23–64. (3x4y)−3\left(\frac{$$3x^4$$}{y}\$$right)^{-3}$$

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 60

1장, 문제 60

Simplify each exponential expression in Exercises 23–64. ${(\frac{3 x^{4}}{y})}^{- 3}$

검증된 단계별 안내

Identify the given expression: $\left(\frac{3x^{4}}{y}\right)^{-3}$.

Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^{n}}$. Apply this to rewrite the expression as $\frac{1}{\left(\frac{3x^{4}}{y}\right)^{3}}$.

Apply the power of a quotient rule: $\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}$. Rewrite the denominator as $\frac{(3x^{4})^{3}}{y^{3}}$.

Rewrite the entire expression as $\frac{1}{\frac{(3x^{4})^{3}}{y^{3}}}$, which is equivalent to multiplying by the reciprocal: $\frac{y^{3}}{(3x^{4})^{3}}$.

Apply the power of a product rule: $(ab)^{n} = a^{n}b^{n}$. Expand the denominator to $3^{3} \cdot (x^{4})^{3}$, and then use the power of a power rule: $(x^{m})^{n} = x^{mn}$ to simplify $x^{4 \cdot 3}$.

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주요 개념

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

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-n) equals 1 divided by x^n. This rule helps simplify expressions by rewriting negative powers as fractions.

Power of a Quotient Rule

When an entire fraction is raised to an exponent, both the numerator and denominator are raised to that power separately. For instance, (a/b)^n equals a^n divided by b^n. This rule is essential for simplifying expressions like (3x^4/y)^-3.

Power of a Product Rule

When a product inside parentheses is raised to an exponent, each factor is raised to that exponent individually. For example, (ab)^n equals a^n times b^n. This helps in breaking down expressions like (3x^4)^-3 for simplification.

Simplify each exponential expression in Exercises 23–64. (3x4y)−3\(\left\)(\(\frac{3x^4}{y}\]\right\))^{-3}