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Chapter 7: Problem 72

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{18 m^{2}}\)

Short Answer

Expert verified
\frac{\frac{5}{3}}{\frac{1}{1}}

Step by step solution

01

Identify the format

Recognize that the expression is in the form of a fractional radicand under a cube root: \ \ \[ \frac{\frac{5}{16}}{3} \]
02

Simplify the cube root of the fraction

Use the property of cube roots that allows splitting the numerator and the denominator: \ \ \[ \frac{\frac{5}{16}}{3} = \frac{\frac{5}{3}}{\frac{16}{3}} \]
03

Simplify the cube root of the numerator

Find the cube root of 5: \ \ \[ \frac{5}{3} = \frac{\frac{1}{5}}{5} \]
04

Simplify the cube root of the denominator

Find the cube root of 16: \ \ \[ \frac{5}{16} = 16^{-3} = 2 \]
05

Combine the results

Write down the simplified form from both the numerator and the denominator: \ \ \[ \frac{\frac{1}{1}}{\frac{2}{2}} = 5 \]

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Key Concepts

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

fractional radicand
A fractional radicand means that the value inside the root is a fraction. In this exercise, the radicand is \(\frac{5}{16}\). Understanding fractional radicands is essential because the numerator and denominator must usually be handled separately when simplifying roots. For instance, with a cube root \(\frac{\frac{5}{16}}{3}\), you need to identify and process both the top (numerator) and bottom (denominator) parts of the fraction independently to simplify effectively.
numerator and denominator properties
When dealing with fractional radicands, applying the properties of roots to both the numerator and the denominator is crucial. Cube roots follow the property: $$\frac{\frac{a}{b}}{c} = \frac{\frac{a}{c}}{\frac{b}{c}}$$ This means you can take the cube root of the numerator and the denominator separately. In our exercise, we have \(\frac{5}{16}\). We need to find the cube root of both 5 and 16. These properties simplify the process of breaking down and solving the problem.
cube root
The concept of the cube root involves finding a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because \(2^3 = 2 * 2 * 2 = 8\). In our problem, we need to find \(\frac{5}{3}\), which means finding both cube root of 5 and the cube root of 16 independently. Recognize the cube root, known as \(\frac{5}{16}\).

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