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Chapter 8: Problem 1

Simplify the following radical expressions by factoring. \(\sqrt{20}\)

Short Answer

Expert verified
2\( \sqrt{5} \)

Step by step solution

01

- Factor the radicand

Identify and factor the number inside the radical. The number 20 can be factored into its prime factors: 20 = 2^2 * 5.
02

- Apply the square root

Apply the square root property to the factored form. Use the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). So, \( \sqrt{20} = \sqrt{2^2 \cdot 5} = \sqrt{2^2} \cdot \sqrt{5} \).
03

- Simplify the square root of perfect squares

Simplify \( \sqrt{2^2} \, \) which equals 2. Now the expression becomes \( 2 \cdot \sqrt{5} \. \)

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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 the process of breaking down a number into its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and itself. For example, 20 can be factored into 2 and 5, where 2 is also a prime number appearing twice.

When prime factorizing a number like 20, start by dividing by the smallest prime number (2) and continue dividing until you're only left with prime numbers:
  • 20 divided by 2 equals 10 (2 is a prime factor)
  • 10 divided by 2 equals 5 (2 is a prime factor)
  • 5 is already a prime number

So, 20 can be written as 2虏 * 5 in its prime factorized form.
Square Root Property
The square root property helps us simplify expressions involving square roots. This property states that the square root of a product is equal to the product of the square roots of each factor. In mathematical form: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).

By using the square root property, we can simplify complex radicals. For instance, if we have \( \sqrt{2^2 \cdot 5} \), we can apply the property to simplify this to \( \sqrt{2^2} \cdot \sqrt{5} \).

This property makes it easier to handle expressions under radicals, especially when one of the factors is a perfect square.
Simplification of Radicals
Simplifying radicals involves writing the radical expression in its simplest form. Here's a step-by-step breakdown of how to simplify radicals using our example: \( \sqrt{20} \).

First, factor the number under the radical using prime factorization: 20 becomes \( 2^2 \cdot 5 \).

Next, we apply the square root property: \( \sqrt{2^2 \cdot 5} = \sqrt{2^2} \cdot \sqrt{5} \).

Lastly, we simplify the square root of the perfect square: \( \sqrt{2^2} \) turns into 2. Therefore, \( \sqrt{20} \) becomes \ 2 \cdot \sqrt{5} \. So, \ \sqrt{20} = 2 \sqrt{5} \ in its simplest form.

This method can be used for other radicals to ensure they are in their most simplified state.

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Most popular questions from this chapter

Solve the following radicals. Be careful. Some of them are undefined. \(\sqrt[5]{-32}\)

Quickly solve these square root problems. \(\sqrt{81}\)

Solve each of the following square root problems by rewriting the radical expression as two separate expressions. \(\sqrt{25 y^{2}}(y>0)\)

In each case, simplify the radical expressions by placing them under the same radical sign. \((\sqrt{(12)})(\sqrt{3})\)

Solve the radical expressions by finding the positive roots. \(\sqrt[4]{16}\)
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