Question

Simplify the radicals. $\sqrt[3]{\frac{-27 a^{4}}{8 a}}$

   Simplify the radicals.
$\sqrt[3]{\frac{-27 a^{4}}{8 a}}$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Julie Miller,Molly… 3rd Edition

Chapter 10, Problem 58

Instant Answer

Solved by Expert Erika Bustos

03/30/2021

Step 1

The fraction $-27a^{4}/8a$ simplifies to $-27a^{3}/8$ because one $a$ in the numerator cancels out with the $a$ in the denominator. $$\sqrt[3]{\frac{-27 a^{4}}{8 a}} = \sqrt[3]{\frac{-27 a^{3}}{8}}$$ Show more…

Show all steps

lock

Thanks for your feedback!

Simplify the radicals. $\sqrt[3]{\frac{-27 a^{4}}{8 a}}$

Play audio

Feedback

Powered by NumerAI

Erika Bustos and 62 other educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Radical Simplification

This involves rewriting an expression under a radical (such as a cube root) in a simpler or more manageable form by applying algebraic identities and properties of exponents. Simplification typically includes reducing coefficients, canceling out common factors, and rewriting expressions to reveal perfect powers that can be extracted from the radical.

Properties of Exponents

Key properties of exponents, such as dividing powers with the same base which results in subtracting their exponents (e.g., a^(m)/a^(n) equals a^(m?n)), are crucial when simplifying expressions under a radical. These properties help simplify algebraic expressions before applying the radical operation.

Cube Roots and Their Properties

Understanding cube roots (and, more generally, odd roots) is essential because they allow negative numbers to remain negative after the root extraction, unlike square roots. The cube root of a perfect cube can be directly taken outside the radical, which simplifies the overall expression.

Simplification of Numerical Fractions within Radicals

Simplifying numerical fractions under the radical is often an important step. This typically involves decomposing numerical coefficients into their prime factors, identifying and extracting perfect cubes (or other relevant powers), and reducing the fraction to its simplest form before applying the radical.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later