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Chapter 9: Problem 205

In the following exercises, simplify. $$ 3 \sqrt{20 x^{2}}-4 \sqrt{45 x^{2}}+5 x \sqrt{80} $$

Short Answer

Expert verified
The simplified expression is \( -6 \sqrt{5x^{2}} + 20x \sqrt{5} \).

Step by step solution

01

- Simplify the Radicals

Simplify each square root term separately. Start by breaking down the numbers inside the radicals into their prime factors.\[ 3 \times \sqrt{20x^{2}} = 3 \times \sqrt{4 \times 5x^{2}} = 3 \times 2 \times \sqrt{5x^{2}} = 6 \sqrt{5x^{2}} \]Next, for the second term:\[ -4 \times \sqrt{45x^{2}} = -4 \times \sqrt{9 \times 5x^{2}} = -4 \times 3 \times \sqrt{5x^{2}} = -12 \sqrt{5x^{2}} \]Finally, for the third term:\[ 5x \sqrt{80} = 5x \sqrt{16 \times 5} = 5x \times 4 \sqrt{5} = 20x \sqrt{5} \]
02

- Combine Like Terms

Combine the terms with like radicals and simplify.The simplified terms from Step 1 are:\[ 6 \sqrt{5x^{2}}, -12 \sqrt{5x^{2}}, 20x \sqrt{5} \]Combine the first two terms as they both contain \( \sqrt{5x^{2}} \) :\[ 6 \sqrt{5x^{2}} - 12 \sqrt{5x^{2}} = -6 \sqrt{5x^{2}} \]
03

- Final Expression

Combine the result with the third term.\[ -6 \sqrt{5x^{2}} + 20x \sqrt{5} \]

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Key Concepts

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

Radical Expressions
A radical expression includes a square root, cube root, or any higher-order root. When simplifying, start by breaking down the numbers inside the radicals into their prime factors. This helps in recognizing perfect squares (or cubes, etc.) which can be taken out of the radical.

For example, in the expression \(3 \sqrt{20x^{2}}\), break down 20 into prime factors: \(4 \times 5\). Since 4 (which is \(2^2\)) is a perfect square, it can be simplified to 2. Hence, \[3 \times \sqrt{20x^{2}} = 3 \times \sqrt{4 \times 5 x^{2}} = 3 \times 2 \times \sqrt{5x^{2}} = 6 \sqrt{5 x^{2}}\]

Continue the same process for the other terms to get a simplified expression.
Like Terms
Like terms in algebra are terms that have the same variables raised to the same power. Only the coefficients (numerical part) of these terms are different. For instance, \(6 \sqrt{5x^{2}}\) and \(-12 \sqrt{5x^{2}}\) are like terms because they both have the same radical part \(\sqrt{5x^{2}}\).

When combining like terms, simply add or subtract their coefficients, keeping the radical part the same. If \(6 \sqrt{5x^{2}} - 12 \sqrt{5x^{2}}\), you get \(-6 \sqrt{5x^{2}}\).

This makes the process of simplifying expressions more straightforward and the resulting expression clearer.
Prime Factorization
Prime factorization is the process of breaking down a number into its prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

For example, the prime factorization of 45 is done by: \[\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}\]

And for 80, it is: \[5x \sqrt{80} = 5x \sqrt{16 \times 5} = 5x \times 4 \sqrt{5} = 20x \sqrt{5}\]

Identifying prime numbers within expressions helps in simplifying radicals by factoring out the square root of perfect squares (or cubes, etc.), simplifying the terms clearer and more manageable.

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Most popular questions from this chapter

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{3}}{\sqrt{m}-\sqrt{5}} $$

In the following exercises, simplify. (a) \(\sqrt[3]{-8}\) (b) \(\sqrt[4]{-81}\) (c) \(\sqrt[5]{-32}\)

In the following exercises, simplify and rationalize the denominator. $$ \frac{4}{9 \sqrt{5}} $$

In the following exercises, check whether the given values are solutions. For the equation \(\sqrt{x+12}=x:\) (a) Is \(x=4 \quad a\) solution? (b) Is \(x=-3\) a solution?

(a) Approximate \(\frac{1}{\sqrt{2}}\) by dividing \(\frac{1}{1.414}\) using long division without a calculator. (b) Rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) gives \(\frac{\sqrt{2}}{2}\). Approximate \(\frac{\sqrt{2}}{2}\) by dividing \(\frac{1.414}{2}\) using long division without a calculator. (C) Do you agree that rationalizing the denominator makes calculations easier? Why or why not?
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