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Chapter 10: Problem 23

Simplify by factoring. $$\sqrt{45}$$

Short Answer

Expert verified
The simplified form of \(\sqrt{45}\) is \(3*\sqrt{5}\)

Step by step solution

01

Factoring the number

First, factor the number 45 into its prime factors. Prime factor of 45 are 3, 3 and 5. Write the prime factors as \(\sqrt{3*3*5}\)
02

Simplify the square root

Now, we simplify the square root by recognizing the square pair 3 * 3. A square pair under a square root simplifies to the base number. Our expression then becomes \(\sqrt{3*3}*5 = 3*\sqrt{5}\).

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Key Concepts

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

Prime Factorization
To simplify square roots using prime factorization, we first need to understand what prime factorization is. Prime factorization is the process of breaking down a composite number into a product of its prime factors. Prime numbers are numbers that are only divisible by 1 and themselves.

For example, the prime factors of 45 are 3, 3, and 5. These are the only prime numbers that, when multiplied together, equal 45 (\(3 \times 3 \times 5 = 45\)). In the context of simplifying square roots, once we have the prime factors, we can easily identify pairs of identical factors, which help us simplify the radical expression.
Radicals
Radicals, often referred to as roots, are expressions that indicate the root of a number. The most common radical is the square root, but there are also cube roots, fourth roots, and so on. The square root of a number is a value that, when multiplied by itself, gives the original number.

Square roots are represented by the radical symbol \( \sqrt{} \) followed by the number whose root we are finding. For instance \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \) Remember, the focus is usually on finding the simplest radical form, which means no square factors remain under the radical symbol.
Square Root Simplification
Square root simplification involves finding an equivalent expression for a square root that has no perfect square factors other than one under the radical. This process is done by using prime factorization to first identify and then remove the square factors.

In our example, \( \sqrt{45} \) simplifies to \( \sqrt{3*3*5} \) because we factored 45 into prime factors. We can then pull out square pairs, such as the pair of 3s in this case, outside the radical. The simplified expression becomes \( 3\sqrt{5} \) This is because \( \sqrt{3*3} = 3 \) and what remains inside the radical is \( \sqrt{5} \) since 5 is a prime number that cannot be broken down further.

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Most popular questions from this chapter

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{5 \sqrt{3}-3 \sqrt{2}}{3 \sqrt{2}-2 \sqrt{3}}$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}+\sqrt{5}}$$

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is \(\frac{w}{h}=\frac{2}{\sqrt{5}-1}\) The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (PICTURE NOT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{35}{5 \sqrt{2}-3 \sqrt{5}}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I raise both sides of an equation to any power, there's always the possibility of extraneous solutions.
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