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Chapter 8: Problem 19

Simplify. $$\sqrt{98}$$

Short Answer

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Step by step solution

01

Identify Perfect Squares

Look for perfect square factors of 98. Notice that 98 can be factored into 49 and 2.
02

Write the Square Root in Terms of Perfect Squares

Express \( \sqrt{98} \) as \( \sqrt{49 \cdot 2} \).
03

Simplify Using Properties of Square Roots

Use the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) to get \( \sqrt{49} \cdot \sqrt{2} \).
04

Calculate the Square Root of the Perfect Square

Calculate \( \sqrt{49} = 7 \), so the expression simplifies to \( 7 \sqrt{2} \).

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Key Concepts

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

perfect squares
A perfect square is a number that is the square of an integer. For example, 1, 4, 9, 16, 25, and 49 are all perfect squares because they can be written as the squares of 1, 2, 3, 4, 5, and 7 respectively.
Identifying perfect squares is crucial when simplifying square roots because it helps in breaking down the number under the square root into more manageable parts.
In the exercise, we identified that 49 is a perfect square that can be used to simplify \( \sqrt{98} \). Understanding perfect squares makes it easier to identify these factors quickly.
properties of square roots
Square roots have specific properties that are helpful when simplifying expressions. One key property is \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This means that the square root of a product is equal to the product of the square roots of the factors.
Using this property, we can simplify complex square root expressions by breaking them up into smaller, easier parts. In the given exercise, we applied this property to \( \sqrt{98} \) by expressing it as \( \sqrt{49 \cdot 2} \).
This allowed us to separate \( \sqrt{49} \) and \( \sqrt{2} \), making the expression easier to work with.
factoring
Factoring is the process of breaking down a number into its prime components or simpler terms that multiply to the original number. Factoring is particularly useful in simplifying square roots.
For the given exercise, we factored 98 into 49 and 2 because 49 is a perfect square. This allowed us to simplify the square root more efficiently.
When factoring, always look for perfect squares among the factors, as that makes simplifying the square root much more straightforward.
Practicing factoring can make it more intuitive which numbers to choose when simplifying square roots, leading to quicker and more accurate results.

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