SME
Grade 9 Math · Problem type
Multiply square roots
Use this for “Simplify √6 × √15.”
Diagram
√
90
=
√
2 × 3 × 3 × 5
.
Each
pair
of
equal
factors
comes
out
as
one:
3.
√
90
=
3
√
10
.
This is an example. Type your own values to start a new problem.
a
×
b
=
p
=
k
r
Formulas
p
=
a
×
b
90 = 6 × 15
k
=
√
largest perfect square factor of
p
3
=
√
largest perfect square factor of 90
r
=
p
k
2
10
=
90
3
2
Assumptions
•
√a × √b = √(ab) for a, b ≥ 0: multiply under one root.
•
Then take out the largest perfect square factor.
Step-by-step
WE KNOW
a = 6, b = 15
FIND
product under one root (p), number outside (k), number left inside (r)
Step 1 · Find product under one root (p)
Use p = a × b
Multiply the numbers under one root.
p
=
6
×
15
p
=
90
Step 2 · Find number outside (k)
Use k = √(largest perfect square factor of p)
Find the largest perfect square that divides the number: each pair of equal prime factors brings one out.
k
=
√
largest perfect square factor of 90
90
=
2
×
3
2
×
5
90 = 9 × 10
√
9
=
3
k
=
3
Step 3 · Find number left inside (r)
Use r = p ÷ k²
What is left under the root: divide out the perfect factor.
r
=
90
3
2
r
=
90
9
r
=
10
→
√
90
=
3
√
10
Check
Put the numbers back in. Both sides should match.
90 = 6 × 15 ✓
3
=
√
largest perfect square factor of 90
✓
10
=
90
3
2
✓
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Exponent properties, rational exponents, simplifying radicals, and rational and irrational numbers
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