Use the quotient property to simplify expressions with higher roots
We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.
Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.
Quotient property of
n Th roots
a
b
n
=
a
n
b
n
and
a
n
b
n
=
a
b
n
when
a
n
and
b
n
are real numbers,
b
≠
0
,
and for any integer
n
≥
2
Simplify:
ⓐ
−108
3
2
3
ⓑ
96
x
7
4
3
x
2
4 .
Solution
ⓐ
−108
3
2
3
Neither radicand is a perfect cube, so use
the Quotient Property to write as one radical.
−108
2
3
Simplify the fraction under the radical.
−54
3
Rewrite the radicand as a product using
perfect cube factors.
(
−3
)
3
·
2
3
Rewrite the radical as the product of two radicals.
(
−3
)
3
3
·
2
3
Simplify.
−3
2
3
ⓑ
96
x
7
4
3
x
2
4
Neither radicand is a perfect fourth power,
so use the Quotient Property to write as one radical.
96
x
7
3
x
2
4
Simplify the fraction under the radical.
32
x
5
4
Rewrite the radicand as a product using
perfect fourth power factors.
2
4
x
4
·
2
x
4
Rewrite the radical as the product of two radicals.
(
2
x
)
4
4
·
2
x
4
Simplify.
2
|
x
|
2
x
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If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.
Simplify:
ⓐ
24
x
7
y
3
3
ⓑ
48
x
10
y
8
4 .
Solution
ⓐ
24
x
7
y
3
3
The fraction in the radicand cannot be
simplified. Use the Quotient Property to
write as two radicals.
24
x
7
3
y
3
3
Rewrite each radicand as a product using
perfect cube factors.
8
x
6
·
3
x
3
y
3
3
Rewrite the numerator as the product of two radicals.
(
2
x
2
)
3
3
3
x
3
y
3
3
Simplify.
2
x
2
3
x
3
y
ⓑ
48
x
10
y
8
4
The fraction in the radicand cannot be
simplified. Use the Quotient Property to
write as two radicals.
48
x
10
4
y
8
4
Rewrite each radicand as a product using
perfect fourth power factors.
16
x
8
·
3
x
2
4
y
8
4
Rewrite the numerator as the product of two radicals.
(
2
x
2
)
4
4
3
x
2
4
(
y
2
)
4
4
Simplify.
2
x
2
3
x
2
4
y
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Add and subtract higher roots
We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of
like radicals
Like radicals
Radicals with the same index and same radicand are called
like radicals
Like radicals have the same index and the same radicand.
9
42
x
4 and
−2
42
x
4 are like radicals.
5
125
x
3 and
6
125
y
3 are not like radicals. The radicands are different.
2
1000
q
5 and
−4
1000
q
4 are not like radicals. The indices are different.
We add and subtract like radicals in the same way we add and subtract like terms. We can add
9
42
x
4
+
(
−2
42
x
4
) and the result is
7
42
x
4 .
