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Chapter 8: Problem 44

Find each product and simplify. $$ \sqrt{9} \cdot \sqrt{50} $$

Short Answer

Expert verified
15 \sqrt{2}

Step by step solution

01

Identify the Square Roots

Recognize the square root terms in the expression: \ \( \sqrt{9} \cdot \sqrt{50} \).
02

Calculate the Square Roots

Evaluate the square roots separately: \ \( \sqrt{9} = 3 \) and \ \( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \).
03

Multiply the Results

Multiply the values obtained in the previous step: \ \( 3 \cdot 5 \sqrt{2} = 15 \sqrt{2} \).

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Key Concepts

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

square roots
Square roots are mathematical operations that allow us to find a number which, when multiplied by itself, gives us the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 is 9. When you see the square root symbol, it looks like this: \( \sqrt{} \) .

Understanding square roots helps simplify many mathematical expressions, particularly when working with square root terms in different equations.
radical expressions
Radical expressions include an expression that contains a square root, cube root, or any other root. In this exercise, \( \sqrt{50} \) is a radical expression, because it involves a square root.

Radical expressions can sometimes be simplified by factoring the number under the root. For instance, \( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \) .

Simplifying helps make the radical expressions easier to work with in further calculations.
multiplication of radicals
When multiplying radicals, we first see if the radicals can be simplified, as shown before. Once simplified, we multiply the terms. In this case, we are given \( \sqrt{9} \cdot\ \sqrt{50} \). First, we simplify: \( \sqrt{9} \) is 3 and \( \sqrt{50} = 5 \sqrt{2} \). Then we multiply those results: \( 3 \cdot\ 5 \sqrt{2} = 15 \sqrt{2} \).

Remember that you can only multiply the terms inside the radicals together or the simplified results, but not mix roots of different types.
simplification
Simplifying the final product of radicals is important to get the answer in its simplest form. Once we multiply \( 3 \) and \( 5 \sqrt{2} \) to get \( 15 \sqrt{2} \), our job isn't quite done until we verify that the expression is in its simplest possible form.

Checking step-by-step ensures no further simplification is possible. This final, simplified result \( 15 \sqrt{2} \) is our answer. Always aim to make your final solution as simple and clear as possible for ease of understanding and further usage.

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Most popular questions from this chapter

Combine like terms. $$ 9 x^{2}+3 x^{2}-2 x+4 x-8+1 $$

To rationalize the denominator of an expression such as \(\frac{4}{\sqrt{3}},\) we multiply both the numerator and denominator by \(\sqrt{3}\). By what number are we actually multiplying the given expression, and what property of real numbers justifies the fact that our result is equal to the given expression?

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{81 m^{4} n^{2}} $$

Find each product and simplify. Simplify the radical \(\sqrt{288}\) in two ways. First, factor 288 as \(144 \cdot 2\) and then simplify. Second, factor 288 as \(48 \cdot 6\) and then simplify. How do the answers compare? Make a conjecture concerning the quickest way to simplify such a radical.

Solve each equation. $$ \sqrt{3 x}+6=x $$
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