Problem 16
Question
Simplify each expression by performing the indicated operation. $$ 6 \sqrt{3 a}+\sqrt{3 a} $$
Step-by-Step Solution
Verified Answer
Answer: The simplified form of the given expression is $$7\sqrt{3a}$$.
1Step 1: Identify the terms to be combined
In the given expression, both terms can be combined as they have the same square root value (3a).
The expression is:
$$
6\sqrt{3a} + \sqrt{3a}
$$
2Step 2: Combine the like terms
Since we have identified both terms as like terms, we can simply combine them by adding their coefficients:
$$
6 \sqrt{3a} + 1 \sqrt{3a} = (6+1)\sqrt{3a}
$$
3Step 3: Simplify the expression
After combining the coefficients, we get:
$$
(6+1)\sqrt{3a} = 7\sqrt{3a}
$$
So the simplified expression is:
$$
7\sqrt{3a}
$$
Key Concepts
Like TermsSquare RootsPerforming OperationsCoefficients
Like Terms
Like terms are an essential concept in algebra. They make simplifying expressions much easier.
Like terms refer to terms in an expression that have the same variable and exponent. For example, in the expression \(6\sqrt{3a} + \sqrt{3a}\), both terms are like terms because they include the same square root \(\sqrt{3a}\).
Identifying like terms allows you to combine them, effectively reducing the complexity of an expression. By adding these terms together, you condense the expression into a simpler, more manageable form. Remember, always check if the terms share identical variable components before combining them.
Like terms refer to terms in an expression that have the same variable and exponent. For example, in the expression \(6\sqrt{3a} + \sqrt{3a}\), both terms are like terms because they include the same square root \(\sqrt{3a}\).
Identifying like terms allows you to combine them, effectively reducing the complexity of an expression. By adding these terms together, you condense the expression into a simpler, more manageable form. Remember, always check if the terms share identical variable components before combining them.
Square Roots
Square roots are a fundamental part of algebra that allow us to simplify expressions involving radicals. The square root of a number \(b\), denoted as \(\sqrt{b}\), is a value that, when multiplied by itself, gives \(b\).
In algebraic expressions like \(6\sqrt{3a} + \sqrt{3a}\), square roots help in recognizing like terms. They also highlight how the terms can be simplified or manipulated. Understanding square roots is crucial, as they frequently appear in mathematical equations and problems.Keep in mind:
In algebraic expressions like \(6\sqrt{3a} + \sqrt{3a}\), square roots help in recognizing like terms. They also highlight how the terms can be simplified or manipulated. Understanding square roots is crucial, as they frequently appear in mathematical equations and problems.Keep in mind:
- Square roots can simplify calculations.
- They can express repeated multiplication in a compact form.
- Understanding them helps in performing various algebraic manipulations.
Performing Operations
Performing operations is about manipulating algebraic expressions to simplify or solve them. This involves combining like terms, factoring, or other methods.
For the expression \(6\sqrt{3a} + \sqrt{3a}\), you add the coefficients of the like terms to simplify it. Here, the operation performed is addition, which combines the terms into one to produce a more straightforward form: \(7\sqrt{3a}\). To perform operations efficiently:
For the expression \(6\sqrt{3a} + \sqrt{3a}\), you add the coefficients of the like terms to simplify it. Here, the operation performed is addition, which combines the terms into one to produce a more straightforward form: \(7\sqrt{3a}\). To perform operations efficiently:
- Identify like terms correctly.
- Use basic arithmetic operations (addition, subtraction, multiplication, division).
- Simplify expressions step by step.
Coefficients
Coefficients are the numerical part of terms in an algebraic expression. They play a vital role when combining like terms and performing operations.
In our expression \(6\sqrt{3a} + \sqrt{3a}\), the coefficients are 6 and 1, respectively. These numbers determine how many times the accompanying variable, or expression, is being considered. By adding these coefficients, you simplify the expression. Understanding coefficients helps in grasping how terms are scaled and combined and is crucial for simplifying expressions effectively. To work with coefficients:
In our expression \(6\sqrt{3a} + \sqrt{3a}\), the coefficients are 6 and 1, respectively. These numbers determine how many times the accompanying variable, or expression, is being considered. By adding these coefficients, you simplify the expression. Understanding coefficients helps in grasping how terms are scaled and combined and is crucial for simplifying expressions effectively. To work with coefficients:
- Ensure terms are like terms before adding or subtracting coefficients.
- Use coefficients to simplify expressions accurately.
- Understand how they affect the overall value of the expression.
Other exercises in this chapter
Problem 15
Simplify each square root. $$ \sqrt{\frac{y^{4}}{x^{3}}} $$
View solution Problem 16
For the following problems, simplify each of the square root expressions. $$ \frac{6-\sqrt{3}}{4+\sqrt{2}} $$
View solution Problem 16
For the following problems, simplify the expressions. $$ \frac{\sqrt{4 x^{2}+4 x+1}}{\sqrt{2 x+1}} $$
View solution Problem 16
For the following problems, solve the square root equations. $$ \sqrt{y+7}=9 $$
View solution