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Chapter 8: Problem 107

Simplify each radical. $$ \sqrt[3]{-\frac{216}{125}} $$

Short Answer

Expert verified
-\frac{6}{5}

Step by step solution

01

- Simplify the Radical by Separating the Numerator and Denominator

Rewrite the cube root of the fraction as the cube root of the numerator divided by the cube root of the denominator: \[ \frac{\root 3 \relax -216}{\root 3 \relax 125} \]
02

- Simplify the Cube Root of the Numerator

Determine the cube root of -216. Since \( -216 = -6^3 \), we find \( \root 3 \relax -216 = -6 \).
03

- Simplify the Cube Root of the Denominator

Determine the cube root of 125. Since \( 125 = 5^3 \), we find \( \root 3 \relax 125 = 5 \).
04

- Combine the Simplified Numerator and Denominator

Combine the results from the previous steps to get the simplified radical: \[ \frac{-6}{5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Roots
A cube root helps us find a number that, when multiplied by itself three times, gives us the original number. For example, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\). Cube roots are critical in simplifying expressions, especially when dealing with fractional values. In the given problem, we start by splitting the fraction inside the cube root into the numerator and the denominator, simplifying each separately.
Radical Simplification
Radical simplification makes expressions easier to work with by reducing them to their simplest form. In our example, simplifying \( \root 3 \relax -216\) and \( \root 3 \relax 125\) involves finding the cube roots of both -216 and 125. By breaking down the fraction into its numerator and denominator, we can directly apply the cube root to each part.
Fraction Simplification
Fraction simplification involves reducing a fraction to its lowest terms. This makes expressions much easier to work with. In our problem, after finding the cube roots individually, we need to simplify \( \frac{-6}{5} \). By simplifying fractions, we ensure we have the simplest possible form of the problem's solution.
Numerator and Denominator
The numerator and denominator are the top and bottom parts of a fraction, respectively. To simplify complex expressions like cube roots of fractions, it's useful to treat them separately. In this example, \(-216\) is the numerator and \(125 \) is the denominator. By simplifying each part on its own, we make the overall fraction much simpler to understand and solve.

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Most popular questions from this chapter

Simplify each radical expression. $$ \sqrt{\frac{9}{100}} $$

Perform each operation and express the answer in simplest form. $$ (\sqrt[3]{2}-1)(\sqrt[3]{4}+3) $$

Solve each equation. (Hint: In Exercises 67 and 68, extend the concepts to fourth root radicals.) $$ \sqrt[3]{3 x^{2}-9 x+8}=\sqrt[3]{x} $$

(a) give the answer as a simplified radical and (b) use a calculator to give the answer correct to the nearest thousandth. The period \(p\) of a pendulum is the time it takes for it to swing from one extreme to the other and back again. The value of \(p\) in seconds is given by $$ p=k \cdot \sqrt{\frac{L}{g}} $$ where \(L\) is the length of the pendulum, \(g\) is the acceleration due to gravity, and \(k\) is a constant. Find the period when \(k=6, L=9 \mathrm{ft},\) and \(g=32 \mathrm{ft}\) per sec \(^{2}\).

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{s^{4}} $$
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