Solving Equations Involving
Square Roots
Finding the Square Root of a Fraction
• You can find the square root of a fraction by
taking the square root of both the numerator
and the denominator
9
16
=
9
16
=
3
4
Checking for Understanding
• Simplify
4
25
Checking for Understanding
• Simplify
49
81
Checking for Understanding
• Simplify
1
16
• We could make estimates if the square roots
are not perfect, but typically we simplify the
square root instead by pulling out perfect
squares so that we are keeping exact values.
• That is a topic for another day.
Discussion
• Complete the pattern:
3x3=9
9=3
5 x 5 = 25
25 = 5
a x a = a²
𝑎² = ___
b x b = b²
𝑏² = ___
c x c = c²
𝑐² = ___
Solving Equations Involving Square
Roots
Remember that when solving algebra problems,
we must preserve the balance in the equation.
We do this by performing inverse (opposite)
operations to both sides of the equation.
x + 3 = 12
5x – 4 = 21
Equations with Roots
𝑥 2 = 16
You should be able to look at this and
immediately know the value for x that makes
the equation true.
We need to be able to prove it with algebra.
Equations with Roots
𝑥 2 = 16
In all equations, we are looking for the value of
1x, in this case the x has been squared. The
inverse of squaring a number is taking the
square root. We will do this to both sides of the
equation.
𝑥 2 = 16
𝑥 2 = 16
x=4
Think Pair Share
• Is there another solution that will satisfy this
equation?
𝑥 2 = 16
𝑥 2 = 16
x=4
YES! (-4) could also be a solution to this equation because (-4) x (-4) = 16
Checking for Understanding
• Solve for x
x² = 4
Checking for Understanding
• Solve for x
49 = x²
Checking for Understanding
• Solve for x
x² = 196
Estimating Non – Perfect Solutions
• You can use the same process for yesterday to
make your estimation
x² = 40
x² = 40
x = 40
40
Estimating Non – Perfect Solutions
• You can use the same process for yesterday to
make your estimation
x² = 27
Solving Equations Involving Cube
Roots
Negative Solutions
• When dealing with square roots, we decided
that we could not take the square root of a
negative number.
• Why can we do this with a cube root?
3
−8 = -2
Lets Do a Few
3
−1 =
3
−64 =
3
−216
Think Pair Share
• When taking the cube root of a negative
number, what must be true about the
solution?
Finding the Cube Root of a Fraction
• You can find the cube root of a fraction by
taking the cube root of both the numerator
and the denominator:
3
3
8
8
=3
27
27
=
2
3
Checking for Understanding
• Simplify
3
1
64
Checking for Understanding
• Simplify
3
−8
125
Checking for Understanding
• Simplify
3
−64
−216
• We could make estimates if the cube roots are
not perfect, but typically we simplify the cube
root instead by pulling out perfect cubes so
that we are keeping exact values.
• That is a topic for another day.
Discussion
• Complete the pattern:
3 x 3 x 3 = 27
3
27 = 3
5 x 5 x 5 = 125
3
125 = 5
a x a x a = a³
b x b x b = b³
c x c x c = c³
3
3
3
𝑎³ = ___
𝑏³= ___
𝑐³= ___
Equations with Roots
𝑥 3 = 216
You should be able to look at this and
immediately know the value for x that makes
the equation true.
We need to be able to prove it with algebra.
Equations with Roots
𝑥 3 = 216
In all equations, we are looking for the value of
1x, in this case the x has been squared. The
inverse of squaring a number is taking the
square root. We will do this to both sides of the
equation.
𝑥 3 = 216
3
3
𝑥³ = 216
x=6
Think Pair Share
• Is there another solution that will satisfy this
equation?
𝑥 3 = 216
3
3
𝑥³ = 216
NO! -6 cubed will produce a negative solution
• Cube roots have only one possible solution,
whereas square roots can have 2.
Checking for Understanding
• Solve for x
x³ = 8
Checking for Understanding
• Solve for x
512 = x³
Checking for Understanding
• Solve for x
x³ = -1000
Estimating Non – Perfect Solutions
• You can use the same process for yesterday to
make your estimation
x³ = 40
3
3
𝑥³= 40
3
x = 40
3
40
Estimating Non – Perfect Solutions
• You can use the same process for yesterday to
make your estimation
x³ = 100