SME
Grade 9 Math · Algebra 1
Exponent properties, rational exponents, simplifying radicals, and rational and irrational numbers
Use this for “Simplify √180.”
Diagram
√
180
=
√
2 × 2 × 3 × 3 × 5
.
Each
pair
of
equal
factors
comes
out
as
one:
2
×
3
=
6.
√
180
=
6
√
5
.
This is an example. Type your own values to start a new problem.
n
=
k
r
Formulas
k
=
√
largest perfect square factor of
n
6
=
√
largest perfect square factor of 180
r
=
n
k
2
5
=
180
6
2
Assumptions
•
√(ab) = √a × √b for a, b ≥ 0.
•
Each pair of equal prime factors comes out as one.
•
The radical is simplest when no square factor but 1 is left inside.
Step-by-step
WE KNOW
n = 180
FIND
number outside (k), number left inside (r)
Step 1 · Find number outside (k)
Use k = √(largest perfect square factor of n)
Find the largest perfect square that divides the number: each pair of equal prime factors brings one out.
k
=
√
largest perfect square factor of 180
180
=
2
2
×
3
2
×
5
180 = 36 × 5
√
36
=
6
k
=
6
Step 2 · Find number left inside (r)
Use r = n ÷ k²
What is left under the root: divide out the perfect factor.
r
=
180
6
2
r
=
180
36
r
=
5
→
√
180
=
6
√
5
Check
Put the numbers back in. Both sides should match.
6
=
√
largest perfect square factor of 180
✓
5
=
180
6
2
✓
Topic check
See what you know: 5 questions on this topic
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Multiply square roots
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Rational or irrational?
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