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Chapter 7: Problem 76

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{256 z^{12}}\)

Short Answer

Expert verified
-\frac{\root 3 \big(6x\big)}{\root 3 \big(y^2\big)}

Step by step solution

01

Identify the components inside the radical

The expression inside the cube root is \(\frac{6x}{y^2}\). Recognize that the cube root applies to the entire fraction.
02

Simplify the cube root

Apply the cube root to both the numerator and the denominator separately: \(-\frac{\root 3 \big(6x\big)}{\root 3 \big(y^2\big)}\).
03

Separate the cube root expressions

Rewrite the expression by taking the cube root of each term: \(-\frac{\root 3 \big(6\big) \times \root 3 \big(x\big)}{\root 3 \big(y^2\big)}\).
04

Simplify the cube roots individually

Keep the cube roots as they are: \(-\frac{\root 3 \big(6\big) \times \root 3 \big(x\big)}{\root 3 \big(y^2\big)}\).
05

Express the final simplified form

Combine the terms to get the final simplified expression: \(-\frac{\root 3 \big(6x\big)}{\root 3 \big(y^2\big)}\).

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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 is essentially the inverse operation of raising a number to the power of three. It helps in finding a number which, when multiplied by itself three times, results in the original number. For example, the cube root of 27 is 3, because \[ 3 \times 3 \times 3 = 27 \. \. \] In our exercise, the cube root applies to \(\frac{6x}{y^2}\), so it can be broken down into two separate cube roots: one for the numerator and one for the denominator. This means: \(\root 3\left( \frac{6x}{y^2} \right) = \frac{\root 3(6x)}{\root 3(y^2)}\.\)
radical simplification
Radical simplification involves breaking down complex expressions containing roots into simpler terms. A key step is to make use of properties of radicals, such as \(\root n(ab) = \root n(a) \times \root n(b)\). In our example, \(\root 3\left( 6x \right)\) can be expressed as \(\root 3(6) \times \root 3(x)\). The idea is to handle each component inside the radical separately to make the overall expression simpler and easier to understand. Likewise, the denominator \(\root 3(y^2)\) stays as it is because \y^2\ has no further simplification under cube root.
numerator and denominator handling
When dealing with fractions under radical signs, it's important to apply the radical separately to both the numerator and the denominator. This maintains the integrity of the fraction while allowing for simpler expressions. In the given exercise: \(-\frac{\root 3 (6x)}{\root 3 (y^2)}\)\, each part is addressed individually; the cube root of 6 and the cube root of x in the numerator, and the cube root of \(\root 3 (y^2)\) in the denominator. By keeping them separate, you simplify the complex radical expression into its most basic form, making it easier to grasp and use in further calculations.

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Most popular questions from this chapter

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{144 x^{3} y^{9}}\)

Work each problem. Replace \(a\) with 3 and \(b\) with 4 to show that, in general, $$ \sqrt{a^{2}+b^{2}} \neq a+b $$

Find the distance between each pair of points. \((\sqrt{2}, \sqrt{6})\) and \((-2 \sqrt{2}, 4 \sqrt{6})\)

Tom owns a condominium in a high rise building on the shore of Lake Michigan, and has a beautiful view of the lake from his window. He discovered that he can find the number of miles to the horizon by multiplying 1.224 by the square root of his eye level in feet from the ground. Use Tom's discovery to do the following. (a) Write a formula that could be used to calculate the distance \(d\) in miles to the horizon from a height \(h\) in feet from the ground. (b) Tom lives on the \(14^{\text {th }}\) floor, which is \(150 \mathrm{ft}\) above the ground. His eyes are \(6 \mathrm{ft}\) above his floor. Use the for- mula from part (a) to calculate the distance, to the nearest tenth of a mile, that Tom can see to the horizon from his condominium window.

In the study of sound, one version of the law of tensions is $$ f_{1}=f_{2} \sqrt{\frac{F_{1}}{F_{2}}} $$ If \(F_{1}=300, F_{2}=60,\) and \(f_{2}=260,\) find \(f_{1}\) to the nearest unit.
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