Question

Simplify the expression. (Assume that $x, y, r, s,$ and $t$ are positive.) $$\left(\frac{x^{3}}{-27 y^{-6}}\right)^{-2 / 3}$$

Name: Problem 45, Chapter 1: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-05-06T00:29:48-08:00
Duration: 1 min 27 s
Channel: Tyler Moulton
Description: Simplify the expression. (Assume that x, y, r, s, and t are positive.) ((x^3)/(-27 y^-6))^-2 / 3

   Simplify the expression. (Assume that $x, y, r, s,$ and $t$ are positive.)
$$\left(\frac{x^{3}}{-27 y^{-6}}\right)^{-2 / 3}$$

Applied Calculus for the MLSS A Brief Approach

Soo T. Tan 10th Edition

Chapter 1, Problem 45

Instant Answer

Solved by Expert Tyler Moulton

05/06/2021

Step 1

We have $\frac{x^{3}}{-27 y^{-6}}$. To simplify this, we can rewrite $-27$ as $(-3)^3$ and $y^{-6}$ as $\left(\frac{1}{y^6}\right)$. So, we have $\frac{x^{3}}{(-3)^3 \cdot \left(\frac{1}{y^6}\right)}$. Simplifying further, we can rewrite $(-3)^3$ as $-27$ Show more…

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Simplify the expression. (Assume that $x, y, r, s,$ and $t$ are positive.) $$\left(\frac{x^{3}}{-27 y^{-6}}\right)^{-2 / 3}$$

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Key Concepts

Exponent Rules

Exponent rules are fundamental guidelines in algebra that dictate how to manipulate expressions involving powers. These include the product of powers rule, quotient of powers rule, and the power of a power rule, all of which are essential for simplifying expressions by systematically reducing and rewriting terms with exponents.

Negative Exponents

Negative exponents indicate that the base should be taken as the reciprocal raised to the corresponding positive exponent. This concept is crucial for rewriting expressions in a more manageable form, especially when combining or simplifying algebraic fractions.

Rational Exponents

Rational exponents combine the ideas of exponentiation and radicals, allowing both operations to be expressed with fractional exponents. They provide a powerful framework for simplifying complex expressions by converting roots to exponent form and applying standard exponent rules.

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