Problem 215
Question
In the following exercises, simplify. $$ 9 \sqrt{2}-8 \sqrt{2} $$
Step-by-Step Solution
Verified Answer
\(\sqrt{2}\)
1Step 1: Identify like terms
The expression contains two terms with the same radical part \(\sqrt{2}\): \ 9\sqrt{2} \ and \ -8\sqrt{2}.\
2Step 2: Combine like terms
Since both terms have the same radical part \(\sqrt{2}\), combine the coefficients \(9\) and \(-8\): \[ 9 - 8 = 1. \] Therefore, \[9\sqrt{2} - 8\sqrt{2} = 1\sqrt{2}.\]
3Step 3: Simplify the expression
The simplified form of \[9\sqrt{2} - 8\sqrt{2}\] is \[\sqrt{2}.\]
Key Concepts
Like TermsRadicalsCoefficients
Like Terms
When you are simplifying expressions, it's crucial to identify like terms. Like terms are terms that have the same variables raised to the same powers, allowing them to be combined. In radical expressions, like terms have the same radical part.
In the example exercise, both terms are like terms because they share the same radical part, \(\sqrt{2}.\). This means we can combine them by adding or subtracting their coefficients. Understanding like terms is essential because it simplifies the process of solving complex expressions.
When you see terms like \(9\sqrt{2}\) and \(-8\sqrt{2}\), you should recognize them as like terms. Practice identifying like terms in various expressions to get more comfortable and efficient with this skill.
In the example exercise, both terms are like terms because they share the same radical part, \(\sqrt{2}.\). This means we can combine them by adding or subtracting their coefficients. Understanding like terms is essential because it simplifies the process of solving complex expressions.
When you see terms like \(9\sqrt{2}\) and \(-8\sqrt{2}\), you should recognize them as like terms. Practice identifying like terms in various expressions to get more comfortable and efficient with this skill.
Radicals
Radicals are symbols used to indicate roots of numbers. The most common radical is the square root, denoted by \(\sqrt{}.\). Simplifying expressions involving radicals requires a good understanding of how to handle these symbols.
For instance, consider the example \(9\sqrt{2} - 8\sqrt{2}.\). Here, \(\sqrt{2}\) is the radical part of both terms. When you simplify such expressions, you focus on the coefficients in front of the radical symbols.
Remember, you can only combine radicals that have the same value under the radical sign. Radicals with different values under the square root symbol, like \(\sqrt{2}\) and \(\sqrt{3},\) are not like terms and cannot be directly combined.
Getting comfortable with radicals will make it easier to tackle more complex algebraic expressions.
For instance, consider the example \(9\sqrt{2} - 8\sqrt{2}.\). Here, \(\sqrt{2}\) is the radical part of both terms. When you simplify such expressions, you focus on the coefficients in front of the radical symbols.
Remember, you can only combine radicals that have the same value under the radical sign. Radicals with different values under the square root symbol, like \(\sqrt{2}\) and \(\sqrt{3},\) are not like terms and cannot be directly combined.
Getting comfortable with radicals will make it easier to tackle more complex algebraic expressions.
Coefficients
Coefficients are numerical or constant factors attached to variables or radical parts in an expression. In the example \(9\sqrt{2} - 8\sqrt{2},\) the numbers 9 and -8 are the coefficients.
When simplifying expressions, you combine like terms by adding or subtracting their coefficients. The radical part remains the same. In the example, you subtract 8 from 9 because both terms have the same \(\sqrt{2}\) radical part:
\[9 - 8 = 1.\]
So, \(9\sqrt{2} - 8\sqrt{2}\) simplifies to \(1\sqrt{2}\) or just \(\sqrt{2}.\)
Understanding how to handle coefficients will help you easily combine terms and simplify expressions properly. Practice with different expressions to become proficient at manipulating coefficients.
When simplifying expressions, you combine like terms by adding or subtracting their coefficients. The radical part remains the same. In the example, you subtract 8 from 9 because both terms have the same \(\sqrt{2}\) radical part:
\[9 - 8 = 1.\]
So, \(9\sqrt{2} - 8\sqrt{2}\) simplifies to \(1\sqrt{2}\) or just \(\sqrt{2}.\)
Understanding how to handle coefficients will help you easily combine terms and simplify expressions properly. Practice with different expressions to become proficient at manipulating coefficients.
Other exercises in this chapter
Problem 213
In the following exercises, simplify. $$ 2 \sqrt{363}-2 \sqrt{300} $$
View solution Problem 214
In the following exercises, simplify. $$ \sqrt{150}+4 \sqrt{6} $$
View solution Problem 216
In the following exercises, simplify. $$ 5 \sqrt{x}-8 \sqrt{y} $$
View solution Problem 217
In the following exercises, simplify. $$ 8 \sqrt{13}-4 \sqrt{13}-3 \sqrt{13} $$
View solution