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Chapter 7: Problem 105

Simplify \(5 \sqrt{12}+3 \sqrt{75}\)

Short Answer

Expert verified
\(25\sqrt{3}\)

Step by step solution

01

Find the prime factorization of 12 and 75

First, we find the prime factorization of both numbers inside the square roots, which are 12 and 75. The prime factorization of a number is the representation of that number as the product of its prime factors. For 12, the prime factors are 2 and 3, so the prime factorization is: \ 12 = 2^2 * 3 \ For 75, the prime factors are 3 and 5, so the prime factorization is: \ 75 = 3 * 5^2
02

Apply the properties of square roots

Now we will apply the properties of square roots to simplify the expression. The general rule for a rational exponent is: \ \(\sqrt[n]{a^n} = a\) if n is a natural number and a is a real number. So, we can simplify \(\sqrt{a^2}\) as a. Let's apply this rule to each square root term: \(5 \sqrt{12} = 5 \sqrt{2^2 * 3} = 5 * 2 \sqrt{3} = 10 \sqrt{3}\) \(3 \sqrt{75} = 3 \sqrt{3 * 5^2} = 3 * 5 \sqrt{3} = 15 \sqrt{3}\)
03

Combine the simplified terms

Now let's combine the simplified expressions: \(10 \sqrt{3} + 15 \sqrt{3}\) Both terms have the same square root, so they can be combined like this: \((10 + 15) \sqrt{3} = 25 \sqrt{3}\) So the final simplified expression is: \(25\sqrt{3}\)

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