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Chapter 7: Problem 71

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{72 k^{2}}\)

Short Answer

Expert verified
\(\frac{\root 3 \relax 4}{\root 3 \relax 9}\)

Step by step solution

01

Understand the Cube Root

The problem requires simplifying the cube root of a fraction. The cube root of a fraction \(\frac{a}{b}\) can be expressed as \(\frac{\root 3 \relax a}{\root 3 \relax b}\).
02

Apply the Cube Root to Numerator and Denominator

Apply the cube root to both the numerator and the denominator: \(\root 3 \relax \frac{4}{9} = \frac{\root 3 \relax 4}{\root 3 \relax 9}\).
03

Simplify the Cube Roots

Find the cube roots of the numbers 4 and 9. Simplifying further is not possible since these cube roots do not simplify to rational numbers. Therefore, the expression remains as \(\frac{\root 3 \relax 4}{\root 3 \relax 9}\).

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Key Concepts

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

Simplifying Radicals
Radicals can seem tricky, but they become easier with practice. Simplifying a radical involves breaking it down into its simplest form. When you encounter a cube root, you want to see if the number under the root can be expressed as a smaller radical or integer. This means checking if the number has factors that are perfect cubes. For example, \(\root 3 \relax 27 = 3\) because \(\root 3 \relax 27 = \root 3 \relax (3^3) = 3\). Unfortunately, some cube roots, like \(\root 3 \relax 4\) or \(\root 3 \relax 9\), don't simplify neatly into rational numbers. Here, both 4 and 9 are not perfect cubes, so we keep them within the radical.
Cube Roots
Cube roots are similar to square roots, but instead of finding the number that produces the original number when squared, you're looking for the number that, when cubed (multiplied by itself twice), gives the original number. For example, \(\root 3 \relax 8 = 2\) because \((2 * 2 * 2 = 8)\). The cube root symbol is \(\root 3 \relax{ }\), and you can apply it to fractions as well as whole numbers. In the case of \(\root 3 \relax{\frac{4}{9}}\), both 4 and 9 stay under their respective cube root symbols, because they can't be simplified further.
Fractions in Algebra
Fractions are an important part of algebra. You often see them combined with other operations, such as radicals. In our example, simplifying \(\root 3 \relax \frac{4}{9}\), we separate the numerator and the denominator and take the cube root of each. Thus, it becomes \(\frac{\root 3 \relax 4}{\root 3 \relax 9}\). Doing this decomposition is crucial because it follows algebraic rules and makes it easier to handle more complex expressions. Remember, breaking down problems into smaller parts is a winning strategy in algebra.
Numerator and Denominator
Understanding numerators and denominators is fundamental in working with fractions. The numerator is the top part, which represents how many parts we have. The denominator is the bottom part, which shows the total number of equal parts the whole is divided into. In the fraction \( \frac{4}{9}\), 4 is the numerator, and 9 is the denominator. When applying the cube root, you handle each part separately. This means finding \(\root 3 \relax 4 \) (the cube root of 4) and \(\root 3 \relax 9\) (the cube root of 9) independently, which gives us \(\frac{\root 3 \relax 4}{\root 3 \relax 9}\).

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Most popular questions from this chapter

Find the equation of a circle satisfying the given conditions. Center: (5,-2)\(;\) radius: 4

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[6]{y^{5}} \cdot \sqrt[3]{y^{2}} $$

Find the equation of a circle satisfying the given conditions. Center: (-4,3)\(;\) radius: 2

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{m-4}{\sqrt{m}+2} $$

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt{\sqrt[3]{\sqrt[4]{x}}} $$
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