§ 7.3
Multiplying and Simplifying Radical
Expressions
Multiplying Radicals
The Product Rule for Radicals
If n a and n b are real numbers, then
n
a  n b  n ab.
The product of two nth roots is the nth root of the product.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.3
Multiplying Radicals
EXAMPLE
4
Multiply: (a) 3 5  3 4 (b) 6 x  5  6 x  5 .
SOLUTION
In each problem, the indices are the same. Thus, we multiply
the radicals by multiplying the radicands.
(a) 3 5  3 4  3 5  4  3 20
(b) 6 x  5  6 x  5  6 x  5x  5  6 x  5
4
4
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.3
5
Multiplying Radicals
Check Point 1 on p 509
Multiply: indices are the same. Note: problems from 1-19.
(a) 5  11
 55
(c) 3 6  3 10
 3 60
(d) 7 2 x  7 6 x3
 7 12x 4
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.3
Simplifying Radicals
EXAMPLE
2 3
Simplify by factoring: (a) 28 (b) 3  32 x y .
SOLUTION
(a) 28  4  7
 4 7
2 7
4 is the greatest perfect square
that is a factor of 28.
Take the square root of each
factor.
Write 4 as 2.
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.3
Simplifying Radicals
CONTINUED
 
(b) 3  32 x 2 y 3  3  8 y 3 4 x 2
 8y 3 is the greatest perfect cube
that is a factor of the radicand.
 3  8 y3  3 4x2
Factor into two radicals.
 2 y 4 x
Take the cube root of  8 y3.
3
2
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.3
Simplifying Radicals
Check Point 2 on p 510 problems from 21-31
(a) 80  16 5
4 5
3
(b) 40
 3 8 5
 23 5
(c) 4 32
 4 16 2
 24 2
(d) 200 x 2 y
 100 x 2  2 y
 10 | x | 2 y
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.3
Simplifying Radicals
EXAMPLE
Simplify:
40x3 .
SOLUTION
We write the radicand as the product of the greatest perfect square factor and another
factor. Because the index of the radical is 2, variables that have exponents that are
divisible by 2 are part of the perfect square factor. We use the greatest exponents that
are divisible by 2.
40x3  4 10 x2  x

4  x 10 x 
2
Use the greatest even power of
each variable.
Group the perfect square factors.
 4x2  10x
Factor into two radicals.
 2 x 10x
Simplify the first radical.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.3
Simplifying Radicals
EXAMPLE
Simplify:
4
96x11 .
SOLUTION
We write the radicand as the product of the greatest 4th power
and another factor. Because the index is 4, variables that have
exponents that are divisible by 4 are part of the perfect 4th factor.
We use the greatest exponents that are divisible by 4.
4
96x11  4 16 6  x8  x3

 
Identify perfect 4th factors.
 4 16 x 8 6 x 3
Group the perfect 4th factors.
 4 16x8  4 6x3
Factor into two radicals.
 2 x 2  4 6 x3
Simplify the first radical.
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.3
Simplifying Radicals
Check Point 4 and 5 on p 512 problems from 39-47
9
(cp4)
(cp5)
11 3
 x 8 y10 z 2  xyz  x4 y5 z xyz
10
 8 x y  5 xy
x y z
3
40 x y
14
3
9
12
2
 2 x 3 y 4 3 5 xy 2
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.3
Simplifying Radicals
Check Point 7 on p 513 problems from 61-77
(a) 6 2
 12
(b) 10 3 16  5 3 2
(c ) 4 4 x 2 y  4 8 x 6 y 3
 4 3
2 3
 503 32  503 8  4
 4 32 x 8 y 4
 50 2 3 4  1003 4
 4 16 x 8 y 4  2
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.3
 2x2 y4 2
DONE
Simplifying Radicals
Simplifying Radical Expressions by Factoring
A radical expression whose index is n is simplified when its radicand
has no factors that are perfect nth powers. To simplify, use the
following procedure:
1) Write the radicand as the product of two factors, one of which is
the greatest perfect nth power.
2) Use the product rule to take the nth root of each factor.
3) Find the nth root of the perfect nth power.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.3
Simplifying Radicals
EXAMPLE
3
If f x   3 48x  2 , express the function, f, in simplified form.
SOLUTION
Begin by factoring the radicand. There is no GCF.
f x   3 48x  2
3
This is the given function.
 3 6  8x  2
3
3
 6  2 x  2
3
 3 6  3 23  3  x  2 
 2x  2 3 6
3
3
Factor 48.
Rewrite 8 as 23.
Take the cube root of each factor.
Take the cube root of 23 and
x  23.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.3
Simplifying Radicals
Simplifying When Variables to Even Powers in
a Radicand are Nonnegative Quantities
For any nonnegative real number a,
n
a n  a.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.3
Multiplying Radicals
EXAMPLE
Multiply and simplify:
(a)
4
4 x 2 y 3 z 3  4 8 x 4 yz 6
(b) 5 8 x 4 y 3 z 3  5 8 xy 9 z 8 .
SOLUTION
(a)
4
4 x 2 y 3 z 3  4 8 x 4 yz 6
 4 4 x 2 y 3 z 3  8 x 4 yz 6
Use the product rule.
 4 32 x 6 y 4 z 9
Multiply.
 4 16  2 x 4 x 2 y 4 z 8 z
Identify perfect 4th factors.


 4 16 x 4 y 4 z 8 2 x 2 z

 4 16 x 4 y 4 z 8  4 2 x 2 z
Group the perfect 4th factors.
Factor into two radicals.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.3
Multiplying Radicals
CONTINUED
 2 xyz2 4 2 x 2 z
Factor into two radicals.
(b) 5 8 x 4 y 3 z 3  5 8 xy 9 z 8
 5 8 x 4 y 3 z 3  8 xy 9 z 8
Use the product rule.
 5 64 x 5 y12 z11
Multiply.
 5 32  2 x 5 y10 y 2 z10 z
Identify perfect 5th factors.


 5 32 x 5 y10 z10 2 y 2 z

Group the perfect 5th factors.
 5 32 x 5 y10 z10  5 2 y 2 z
Factor into two radicals.
 2 xy 2 z 2
Simplify the first radical.
5
2 y2z
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.3
Simplifying Radicals
Important to Remember:
A radical expression of index n is not simplified if you can take any
roots – that is if there are any factors of the radicand that are perfect
nth powers.
Take all roots.
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.3