Square Roots and Cube Roots
May 7, 2012
Square Root
• The square root of a nonnegative number is a
number that, when multiplied by itself, is equal
to that nonnegative number.
• Square roots have positive and negative
solutions.
• The symbol
represents a square root.
• Taking the square root of a number is the
opposite, or inverse, of squaring the number.
Solving Square Roots
EXAMPLE #1
You can solve an equation by taking the square
root of both sides.
𝑥 2 = 100
𝑥 2 = 100
x
x
= 100 ⟶ 10 x 10 = 100 and -10 x -10 = 100
= ± 10
Solving Square Roots
EXAMPLE #2
The number under the square root symbol is called the
radicand. If the radicand is not a perfect square, the
square root is irrational.
𝑥2 = 5
𝑥 2 = 5 ⟶ 5 is not a perfect square
x
= 5
x
=± 5
x
= ± 2.23606…
Cube Root
• To find the cube root of a number, find the
number that, when multiplied by itself two times
(a total of three factors), is equal to that number.
• The symbol
represents a cube root.
• Taking the cube root of a number is the opposite,
or inverse, of cubing a number (raising a number
to the third power).
Solving Cube Roots
EXAMPLE #1
You can solve an equation by taking the cube root
of both sides.
𝑥 3 = 125
3
x
x
𝑥3
3
= 125
3
= 125 ⟶ 5 x 5 x 5 = 125
=5
Practice Problem 1
Solve for x
𝑥 2 = 169
SOLUTION
Take the square root of both sides of the equation.
𝑥 2 = 169
Find the value of each square root.
𝑥2
= 𝑥 ⟶ x • x = 𝑥2
169 = 13 ⟶ 13 • 13 = 169
x = ±13
Practice Problem 2
Approximate the value of x
SOLUTION
𝑥 2 = 50
Take the square root of both sides of the equation.
𝑥 2 = 50
Find or approximate the value of each square root.
𝑥2
= 𝑥 ⟶ x • x = 𝑥2
50 ≈ ±7.1
x≈ ±7.1
Practice Problem 3
Solve for x
𝑥 3 = 216
SOLUTION
Take the cube root of both sides of the equation.
3
𝑥3
3
= 216
Find the value of each cube root.
3
3
𝑥3
216
x=6
= 𝑥 ⟶ x • x • x = 𝑥3
= 6 ⟶ 6 • 6 • 6 = 216