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§ 7.4
Adding, Subtracting, and Dividing Radical Expressions
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.4
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 elephantsbut
5 tigers + 3 gorillas = ???
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.4
Combining Radicals
EXAMPLEEXAMPLE
Simplify (add or subtract) by combining like radical terms: .57276(b)7479(a) 3333 xx
SOLUTIONSOLUTION
Apply the distributive property.
Simplify.
33 7479(a) 3 749
3 7533 57276(b) xx
33 57276 xx
Apply the distributive property.
Group like terms.
351726 x
Simplify.3478 x
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.4
Combining Radicals
EXAMPLEEXAMPLE
Simplify by combining like radical terms, if possible:
.54(b)3124(a) 3 23 24 xyxyxxx
SOLUTIONSOLUTION
Factor the radicands using the greatest perfect square factors.
Apply the distributive property.
Take the square root of each factor.
3124(a) xx 334 xx
334 xx
332 xx 24
312 x
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.4
Combining Radicals
3 23 24 54(b) xyxyx Factor the radicands using the greatest perfect cube factors.
Simplify.
Take the cube root of each factor.
33 x
CONTINUECONTINUEDD
3 23 23 54 xyxxyx
3 23 23 3 54 xyxxyx
Apply the distributive property.
3 23 2 54 xyxxyx xx 3 3
3 254 xyxx
Simplify.3 29 xyx
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.4
Simplifying Radicals
The Quotient Rule for RadicalsIf and are real numbers and , then
The nth root of a quotient is the quotient of the nth roots.
n a n b 0b
.n
n
n
b
a
b
a
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.4
Simplifying Radicals
EXAMPLEEXAMPLE
Simplify using the quotient rule: .13
(b)81
50(a) 4
12
7
8
3
x
y
y
x
SOLUTIONSOLUTION
44
2
4
2
8
3
8
3
9
25
9
225
9
225
81
50
81
50(a)
y
xx
y
xx
y
xx
y
x
y
x
3
4 3
3
4 34 4
3
4 34
4 12
4 7
412
7 1313131313(b)
x
yy
x
yy
x
yy
x
y
x
y
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.4
Combining Radicals
Dividing Radical ExpressionsIf and are real numbers and , then
To divide two radical expressions with the same index, divide the radicands and retain the common index.
n a n b 0b
.nn
n
b
a
b
a
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.4
Combining Radicals
EXAMPLEEXAMPLE
Divide and, if possible, simplify: .2
250(b)
3
54(a)
3 3
3 35
24
117
x
yx
ba
ba
SOLUTIONSOLUTION
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.
24
117
24
117
3
54
3
54(a)
ba
ba
ba
ba
2114718 ba
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
Combining Radicals
33
35
3 3
3 35
2
250
2
250(b)
x
yx
x
yx
131118 ba
CONTINUECONTINUEDD Simplify.
abba 29 1210Factor using the greatest perfect square factor.
abba 29 1210 Factor into two radicals.
Simplify.abba 23 65
3 335125 yx
Divide the radicands and retain the common index.
Divide factors in the radicand. Subtract exponents on common bases.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.4
Combining Radicals
CONTINUECONTINUEDD Simplify.
Factor using the greatest perfect square factor.
Factor into two radicals.
Simplify.
3 32125 yx
3 23125 xy
3 23 3125 xy 3 25 xy
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.4
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 youhave like radicals that can be combined.
37
3532
7512
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