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Chapter 5: Problem 23

Simplify. $$ \sqrt{162} $$

Short Answer

Expert verified
Simplified form is \( 9 \sqrt{2} \).

Step by step solution

01

Identify perfect square factors

First, identify the factors of 162 that include a perfect square. Notice that 162 can be factored as 81 and 2, and 81 is a perfect square.
02

Express inside of radical

Rewrite the square root of 162 using its factors: \ \( \sqrt{162} = \sqrt{81 \cdot 2} \ \)
03

Simplify the square root

Simplify using the property of square roots \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) : \ \( \sqrt{81 \cdot 2} = \sqrt{81} \cdot \sqrt{2} \ \)
04

Calculate the square root of the perfect square

Since 81 is a perfect square and \( \sqrt{81} = 9 \) : \ \( \sqrt{81} \cdot \sqrt{2} = 9\sqrt{2} \ \)

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Key Concepts

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

Perfect Square Factors
To simplify square roots, it's important to identify perfect square factors within the number under the radical. A perfect square is an integer that is the square of another integer, like 4, 9, 16, and 81. When we break down a number like 162, we look for factors that include these perfect squares. For 162, we can factor it into 81 and 2, where 81 is a perfect square because it is equal to 9 squared. This step is crucial for simplifying because it allows us to separate the number into a part that can be simplified and another that might remain under the radical.
Properties of Square Roots
After identifying the perfect square factors, we use properties of square roots to simplify the expression. One key property is that the square root of a product is equal to the product of the square roots: \[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \] Applying this, we rewrite \ \( \sqrt{162} = \sqrt{81 \cdot 2} \ \). Next, we separate the factors under the radical: \ \( \sqrt{81} \cdot \sqrt{2} \ \). This property helps simplify by isolating the square root of the perfect square, which can be easily calculated.
Radical Expressions
A radical expression involves a root symbol, usually the square root. Simplifying such expressions is easier when you know the properties of radicals and can identify perfect squares. For \ \( \sqrt{162} \ \) , we saw that 162 can be written as 81 times 2. Since \ \( \sqrt{81} \ = \ 9 \ \), \ \( \9 \sqrt{2} \ \) is the simplified form. The goal is to make the radical simpler while keeping it mathematically equivalent. In general, simplifying radicals makes calculations easier and expressions clearer.

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