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Chapter 10: Problem 13

Perform the indicated operations. Simplify all answers as completely as possible. Assume that all variables appearing under radical signs are nonnegative. $$\sqrt{18} \sqrt{32}$$

Short Answer

Expert verified
24

Step by step solution

01

- Multiply the Radicals

Multiply the two square roots together: \ \ \( \sqrt{18} \sqrt{32} \) results in \ \( \sqrt{18 \cdot 32} \).
02

- Simplify the Product Under the Radical

Calculate the product inside the radical: \ \( 18 \cdot 32 = 576 \). So \ \( \sqrt{576} \).
03

- Simplify the Radical

Simplify \ \( \sqrt{576} \) by recognizing that 576 is a perfect square: \ \( \sqrt{576} = 24 \).

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Key Concepts

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

radical multiplication
Radical multiplication involves multiplying numbers that are under a square root. For example, in the exercise \(\text{ \sqrt{18} \sqrt{32}} \), when you multiply these two radicals, you get \( \sqrt{18} \times \sqrt{32} = \sqrt{576} \). \ The key is to understand that the product of two square roots is the square root of the product of the numbers.
perfect squares
A perfect square is a number that can be expressed as the product of an integer with itself. For instance, 49 is a perfect square because it can be written as 7 times 7 (i.e., \( 7 \times 7 = 49 \)). \ Recognizing perfect squares helps in simplifying radicals easily. In the given example, 576 is recognized as a perfect square because \( 24 \times 24 = 576 \), making \( \sqrt{576} = 24 \).
square roots
Understanding square roots is fundamental when working with radicals. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). \ \ To simplify a square root, always look for factors of the number that are perfect squares. In the problem \( \sqrt{576}\), since 576 is a product of 24 multiplied by itself, simplifying yields 24.

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Most popular questions from this chapter

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$4 \sqrt{96}-5 \sqrt{24}$$

Rationalize the denominators and simplify. $$\frac{8}{\sqrt{5}-\sqrt{3}}-\frac{12}{\sqrt{3}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{27}+\frac{4}{\sqrt{3}}$$

Use a calculator to find \(\sqrt{80} .\) Then simplify \(\sqrt{80}\) and again use the calculator to compute the value of the simplified form. Are the results the same? Should they be?

Factor the given expression as completely as possible. $$2 x^{2}-50$$
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