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Chapter 9: Problem 7

Simplify the radical expression. $$\sqrt{45}$$

Short Answer

Expert verified
The simplified radical expression is \(3\sqrt{5}\).

Step by step solution

01

Prime Factorization

Calculate the prime factorization of 45. In other words, break down the number into its basic multiplicative building blocks, which are prime numbers. The prime factorization for 45 is \(3^{2}*5\).
02

Application of the Law of Exponents in Radicals

Apply the fact that the square root of \(x^{2}\) equals x. Here, in the expression, there are two 3's, so one 3 can be taken out of the radical, and the 5 remains under the radical.
03

Simplification

Combine the numbers that were taken out of the radical and the ones that remained inside it. The resulting expression is \(3\sqrt{5}\).

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Key Concepts

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

Prime Factorization
Prime factorization is like "pulling apart" a number until you can go no further using just prime numbers. Prime numbers are numbers greater than 1 that cannot be divided by any other number except 1 and themselves. They are the basic building blocks of all numbers.
To prime factorize a number, you continually divide the number by its smallest prime factor until you end up with the number 1. For example, to perform prime factorization on 45:
  • 45 is divisible by the smallest prime number, 3.
  • Divide 45 by 3 to get 15: \(45 \div 3 = 15\)
  • 15 is again divisible by 3.
  • Divide 15 by 3 to get 5: \(15 \div 3 = 5\)
  • 5 is a prime number, so you stop here.
This gives the prime factorization of 45 as \(3^{2} \times 5\). Performing prime factorization is essential when simplifying radicals because it helps you to identify perfect squares.
Law of Exponents
The law of exponents provides rules on how to handle powers or indices when multiplying or dividing expressions with exponents. One crucial rule for simplifying square roots is that the square root of a squared number is simply the base of the exponent: \(\sqrt{x^2} = x\).
Here's how it applies: when you have prime factored 45 as \(3^{2} \times 5\):
  • Identify pairs: The 3's form a perfect square, \(3^2\).
  • Apply the square root: \(\sqrt{3^2} = 3\).
  • This allows you to "take out" one 3 from under the square root.
It's important because it allows us to simplify radical expressions by reducing the number inside the radical as much as possible. Recognizing these perfect pairs is key to simplifying radicals effectively.
Square Roots
A square root essentially "asks" what number multiplied by itself will give you the original number under the root. For example, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).
When simplifying a square root, like \(\sqrt{45}\), you often try to express the number as a product of perfect squares, making it easier to "free" some of the numbers from the square root. This is why prime factorization is so helpful.
After you've expressed \(45\) as \(3^2 \times 5\), recognizing \(3^2\) as a perfect square allows you to simplify \(\sqrt{45}\) to \(3\sqrt{5}\).
  • The 3 comes outside the radical since \(3^2\) is a perfect square.
  • The 5 stays inside, as it is not a perfect square.
This results in a simplified form of the radical which is shorter and more manageable.

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Most popular questions from this chapter

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