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Chapter 10: Problem 30

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

Short Answer

Expert verified
The simplified form of \sqrt{45} is 3\sqrt{5}.

Step by step solution

01

Express the number under the square root as a product of prime factors

Break 45 down into its prime factors. We can write 45 as 9 multiplied by 5, so: \[ 45 = 9 \times 5 \]
02

Simplify using the property of square roots

We know \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] Use this property to rewrite \sqrt{45} as: \[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} \]
03

Evaluate the square root of the perfect square

The square root of 9 is 3. So, we get: \[ \sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5} \]

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Key Concepts

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

Factoring
Factoring is the process of breaking down numbers into smaller components that, when multiplied together, yield the original number. In the context of square roots, factoring helps simplify the expression under the root. Let's start with \( \sqrt{45} \). The first step is to express 45 as the product of smaller numbers. Here, 45 can be written as 9 multiplied by 5. This is because 9 \(\times \) 5 equals 45. So, we have: \[ 45 = 9 \times 5 \] Factoring makes it easier to apply mathematical properties to simplify our problem.
Prime Factors
Prime factors are the prime numbers that, when multiplied together, yield the original number. A prime number is a number that has no other divisors than 1 and itself. In simplifying square roots, it's often helpful to break numbers down into their prime factors. For example, the number 45 can be broken down into its prime factors: \[ 45 = 3 \times 3 \times 5 \] This is because 9 can be further factored into 3 \(\times \) 3. By identifying the prime factors, we can more easily simplify the square root expression.
Property of Square Roots
The property of square roots states that the square root of a product is equal to the product of the square roots of each factor. Mathematically, this is written as: \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] Using this property helps to break down complex square root expressions into more manageable parts. For instance, with \( \sqrt{45} \), we use our previous factorization to rewrite it as: \[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} \] By breaking the expression into these simpler components, we can easily proceed to the next step.
Perfect Square
A perfect square is a number that is the square of an integer. Recognizing perfect squares helps to simplify square root problems quickly. From our example, 9 is a perfect square because it is equal to 3 \times 3. So, the square root of 9 is simply 3: \[ \sqrt{9} = 3 \] Taking our expression, \(\sqrt{45} \), we now substitute back in: \[ \sqrt{45} = \sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5} \] This results in the simplified form: 3 \sqrt{5}. Recognizing and extracting perfect squares within the original number significantly simplifies the overall calculation.

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Most popular questions from this chapter

The absolute value of a complex number \(a+b i\) is its distance from the origin. Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}} .\) Find the absolute value of each complex number. $$ |3+4 i| $$

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