1. § 7.4
Adding, Subtracting, and Dividing Radical Expressions

2. 2 elephants + 3 elephants = 5 elephants
Combining Radicals
Apples to apples, oranges to oranges,… you can only add “like” things….
Two or more radical expressions that have the same indices and the same radicands are called like radicals.
Like radicals can be combined under addition in exactly the same way that we combined like terms under addition. Examples of this process follow.
2 elephants + 3 elephants = 5 elephants
but
5 tigers + 3 gorillas = ???
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.4

3. Combining Radicals
EXAMPLE
Simplify (add or subtract) by combining like radical terms:
SOLUTION
Apply the distributive property.
Simplify.
Group like terms.
Apply the distributive property.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.4

4. Combining Radicals
EXAMPLE
Simplify by combining like radical terms, if possible:
SOLUTION
Factor the radicands using the greatest perfect square factors.
Take the square root of each factor.
Apply the distributive property.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.4

5. Combining Radicals Simplify.
CONTINUED
Simplify.
Factor the radicands using the greatest perfect cube factors.
Take the cube root of each factor.
Apply the distributive property.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.4

6. The Quotient Rule for Radicals
Simplifying Radicals
The Quotient Rule for Radicals
If and are real numbers and , then
The nth root of a quotient is the quotient of the nth roots.
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.4

7. Simplifying Radicals Simplify using the quotient rule: EXAMPLE
SOLUTION
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.4

8. Dividing Radical Expressions
Combining Radicals
Dividing Radical Expressions
If and are real numbers and , then
To divide two radical expressions with the same index, divide the radicands and retain the common index.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.4

9. Combining Radicals Divide and, if possible, simplify:
EXAMPLE
Divide and, if possible, simplify:
SOLUTION
In each part of this problem, the indices in the numerator and the denominator are the same. Perform each division by dividing the radicands and retaining the common index.
Divide the radicands and retain the common index.
Divide factors in the radicand. Subtract exponents on common bases.
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.4

10. Combining Radicals Simplify.
CONTINUED
Simplify.
Factor using the greatest perfect square factor.
Factor into two radicals.
Simplify.
Divide the radicands and retain the common index.
Divide factors in the radicand. Subtract exponents on common bases.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.4

11. Combining Radicals Simplify.
CONTINUED
Simplify.
Factor using the greatest perfect square factor.
Factor into two radicals.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.4

12. Combining Radicals Important to remember:
Like radicals have the same indices and radicands. Like radicals can be added or subtracted using the distributive property.
In some cases, you cannot see that radicals are “like” until you simplify them. When attempting to combine radicals, you should simplify the radicals first. Then you may see that you
have like radicals that can be combined.
Are we like? You don’t look like me.
Yep. I’m 2 square roots of 3 and you are 5 square roots of 3.
We have the same indices and radicands. We’re like!
Let’s see…2 of them + 5 of them = 7 of them
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.4