Raising a Product or Quotient
to a Power
Andrew Gloag
Eve Rawley
Anne Gloag
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Printed: April 30, 2015
AUTHORS
Andrew Gloag
Eve Rawley
Anne Gloag
www.ck12.org
C HAPTER
Chapter 1. Raising a Product or Quotient to a Power
1
Raising a Product or
Quotient to a Power
Here you’ll learn how to use the product and quotient properties to evaluate radical expressions. You’ll also use
those properties to write radicals in simplest radical form.
p
What if you had a radical expression like 50x3 y5 ? How could you simplify this expression? After completing this
Concept, you’ll be able to use the product and quotient properties of radicals to write them in simplest form.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/133292
CK-12 Foundation: Properties of Radicals
Guidance
A radical reverses the operation of raising a number to a power. For example, the square of 4 is 42 = 4 · 4 = 16, and
√
so the square root of 16 is 4. The symbol for a square root is
. This symbol is also called the radical sign.
In addition to square roots, we can also take cube roots, fourth roots, and so on. For example, since 64 is the cube of
4, 4 is the cube root of 64.
√
3
64 = 4
since
43 = 4 · 4 · 4 = 64
We put an index number in the top left corner of the radical sign to show which root of the number we are seeking.
Square roots have an index of 2, but we usually don’t bother to write that out.
√
√
2
36 = 36 = 6
The cube root of a number gives a number which when raised to the power three gives the number under the radical
sign. The fourth root of number gives a number which when raised to the power four gives the number under the
radical sign:
√
4
81 = 3
since
34 = 3 · 3 · 3 · 3 = 81
And so on for any power we can name.
1
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Even and Odd Roots
Radical expressions that have even indices are called even roots and radical expressions that have odd indices are
called odd roots. There is a very important difference between even and odd roots, because they give drastically
different results when the number inside the radical sign is negative.
Any real number raised to an even power results in a positive answer. Therefore, when the index of a radical is even,
the number inside the radical sign must be non-negative in order to get a real answer.
On the other hand, a positive number raised to an odd power is positive and a negative number raised to an odd power
is negative. Thus, a negative number inside the radical sign is not a problem. It just results in a negative answer.
Example A
Evaluate each radical expression.
√
a) 121
√
3
b) 125
√
4
c) −625
√
5
d) −32
Solution
√
a) 121 = 11
√
3
b) 125 = 5
√
3
c) −625 is not a real number
√
5
d) −32 = −2
Use the Product and Quotient Properties of Radicals
Radicals can be re-written as rational powers. The radical:
√
n
m
an is defined as a m .
Example B
Write each expression as an exponent with a rational value for the exponent.
√
a) 5
√
b) 4 a
p
c) 3 4xy
√
6
d) x5
Solution
√
1
a) 5 = 5 2
√
1
b) 4 a = a 4
p
1
c) 3 4xy = (4xy) 3
√
6
5
d) x5 = x 6
As a result of this property, for any non-negative number a we know that
√
n
n
an = a n = a.
Since roots of numbers can be treated as powers, we can use exponent rules to simplify and evaluate radical
expressions. Let’s review the product and quotient rule of exponents.
2
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Chapter 1. Raising a Product or Quotient to a Power
Raising a product to a power:
Raising a quotient to a power:
(x · y)n = xn · yn
n
x
xn
= n
y
y
In radical notation, these properties are written as
Raising a product to a power:
Raising a quotient to a power:
√ √
√
m
x·y = m x · m y
√
r
m
x
x
m
= √
m
y
y
A very important application of these rules is reducing a radical expression to its simplest form. This means that we
apply the root on all the factors of the number that are perfect roots and leave all factors that are not perfect roots
inside the radical sign.
√
For example, in the expression 16, the number 16 is a perfect square because 16 = 42 . This means that we can
simplify it as follows:
√
√
16 = 42 = 4
Thus, the square root disappears completely.
√
On the other hand, in the expression 32, the number 32 is not a perfect square, so we can’t just remove the square
root. However, we notice that 32 = 16 · 2, so we can write 32 as the product of a perfect square and another number.
Thus,
√
√
32 = 16 · 2
If we apply the “raising a product to a power” rule we get:
√
√
√
√
32 = 16 · 2 = 16 · 2
Since
√
√
√
√
16 = 4, we get: 32 = 4 · 2 = 4 2
Example C
Write the following expressions in the simplest radical form.
√
a) 8
√
b) 50
r
125
c)
72
Solution
The strategy is to write the number under the square root as the product of a perfect square and another number. The
goal is to find the highest perfect square possible; if we don’t find it right away, we just repeat the procedure until
we can’t simplify any longer.
3
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a)
√
√
8 = 4 · 2.
√ √
4 · 2.
√
2 2.
We can write 8 = 4 · 2, so
With the “Raising a product to a power” rule, that becomes
√
Evaluate 4 and we’re left with
b)
√
√
50 = 25 · 2
√
√
√
= 25 · 2 = 5 2
We can write 50 = 25 · 2, so:
Use “Raising a product to a power” rule:
c)
r
Use “Raising a quotient to a power” rule to separate the fraction:
Re-write each radical as a product of a perfect square and another number:
√
125
125
= √
72
72
√
√
25 · 5 5 5
= √
= √
36 · 2 6 2
The same method can be applied to reduce radicals of different indices to their simplest form.
Example D
Write the following expression in the simplest radical form.
√
3
a) 40
r
4 162
b)
80
√
3
c) 135
Solution
In these cases we look for the highest possible perfect cube, fourth power, etc. as indicated by the index of the
radical.
√
√
3
3
a)
Here
we are√looking for the product of the highest perfect cube and another number. We write: 40 = 8 · 5 =
√
√
3
3
3
8 · 5=2 5
b) Here we are looking for the product of the highest perfect fourth power and another number.
r
Re-write as the quotient of two radicals:
Simplify each radical separately:
Recombine the fraction under one radical sign:
4
√
4
162
162
= √
4
80
80
√
√
4
4
81 · 2
81
= √
= √
4
4
16 · 5
16
r
3 4 2
=
2 5
√
√
4
4
· 2 3 2
√
= √
4
4
· 5 2 5
c) Here we are looking for the product of the highest perfect cube root and another number. Often it’s not very easy
to identify the perfect root in the expression under the radical sign. In this case, we can factor the number under the
radical sign completely by using a factor tree:
4
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Chapter 1. Raising a Product or Quotient to a Power
We see that 135 = 3 · 3 · 3 · 5 = 33 · 5. Therefore
√
√
√
√
√
3
3
3
3
3
135 = 33 · 5 = 33 · 5 = 3 5.
(You can find a useful tool for creating factor trees at http://www.softschools.com/math/factors/factor_tree/ . Click
on “User Number” to type in your own number to factor, or just click “New Number” for a random number if you
want more practice factoring.)
Now let’s see some examples involving variables.
Watch this video for help with the Examples above.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/133293
CK-12 Foundation: Properties of Radicals
Vocabulary
• The symbol for a square root is
√
. This symbol is also called the radical sign.
Guided Practice
Write the following expression in the simplest radical form.
p
a) 12x3 y5
s
7
4 1250x
b)
405y9
Solution
Treat constants and each variable separately and write each expression as the products of a perfect power as indicated
by the index of the radical and another number.
a)
p
p
√
√
12x3 y5 = 12 · x3 · y5
√
√
p
√ √ √ Simplify each radical separately:
4·3 ·
x2 · x ·
y4 · y = 2 3 · x x · y2 y
p
Combine all terms outside and inside the radical sign:
= 2xy2 3xy
Re-write as a product of radicals:
5
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b)
s
Re-write as a quotient of radicals:
Simplify each radical separately:
Recombine fraction under one radical sign:
4
√
4
1250x7
1250x7
p
=
4
405y9
405y9
√
√
√
√
√
4
4
4
4
4
5 2 · x · x3
5x 2x3
625 · 2 · x4 · x3
√
p
√
= √
= 4
√ =
4
4
81 · 5 · 4 y4 · y4 · y 3 5 · y · y · 4 y 3y2 5y
s
5x 4 2x3
= 2
3y
5y
Explore More
Evaluate each radical expression.
√
1. √ 169
4
2. √−81
3
3. √−125
5
4. 1024
Write each expression as a rational exponent.
√
3
5. 14
√
4
6. √
zw
7. pa
9
8.
y3
Write the following expressions in simplest radical form.
√
9. √24
10. √ 300
5
11. r96
240
12.
567
√
3
13. √500
6
14. √64x8
3
15. s48a3 b7
16x5
3
16.
135y4
6