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7.1 Day 1 - Nth Roots
Prior Knowledge - Square Roots
2 is a square root of 4 because 22 = 4
and
−2 is a square root of 4 because (−2)2 = 4
1) _____ and _____ are a square roots of 16 because _______ = 16 and _______ = 16.
2) 5 is a square root of ______ because 52 = ________ .
3) _____ is another square root of _____ because:
* We can extend the idea of square roots to other types of root *
2 is a cube root of 8 because 23 = 8
and
–2 is a cube root of –8 because (−2)3 = −8
1) _____ is a cube root of 27 because _______ = 27
2) –4 is a cube root of _______ because (−4)3 = _________ .
3 is a fourth root of 81 because 34 = 81 and –5 is a fourth root of 625 because (−5)4 = 625
3) _____ and _____ are both fourth roots of 1 because _______ = 1 and _______ = 1.
4) –2 is a fourth root of _______ because (−2)4 = _________ .
5) _____ is another fourth root of _____ because:
In General, for an integer n greater than 1, if 𝑏 𝑛 = 𝑎, then b is an nth root of a.
𝑛
An nth root of a is written as √𝑎, where n is the index of the radical.
REAL Nth Roots
Let n be an integer greater than1 and let a be a real number.
If n is odd,
If n is even and a > 0,
If n is even and a = 0,
If n is even and a < 0,
then a has ______
then a has ______
then a has ______
then a has ______
real nth root(s).
real nth root(s).
real nth root(s).
real nth root(s).
4
4
4
Examples:
3
√8 = 2
3
√−27 = −3
√16 = ±2
2
√25 = ±5
√0 = 0
2
√0 = 0
√−81 =
No Real numbers
Practice – Find the indicated nth root(s) of a.
1) n = 3, a = –125
2) n = 4, a = 16
3) n = 5, a = –32
5) n = 5, a = –1
6) n = 7, a = 0
n is odd, so 1 real root.
3
√−125 = −5
4) n = 3, a = 64
To solve simple equations involving 𝑥 𝑛 , isolate the power and take the nth root of each side.
Example: (𝑥 − 2)3 − 10 = 0
(𝑥 − 2)3 = 10
Isolate the power
3
3
√(𝑥 − 2)3 = √10
3
𝑥 − 2 = √10
3
𝑥 = √10 + 2 ≈ 4.15
Take the cube root of both sides
Continue to solve
Find an approximation, if necessary, using your calculator
1) 2𝑥 4 = 162
2) 6𝑥 4 = 3750
3) (𝑥 + 1)3 = 18
4) 5𝑦 4 = 80
5) (𝑦 − 1)3 = 32
6) 2𝑥 6 + 12 = 8