Goal: Pull Perfect Squares Out of Radicals

Tiny Concept

Sometimes the number inside a radical is not a perfect square.

Example:

$\sqrt{72}$

72 is not on our perfect square list.

So instead, we look for the largest perfect square factor hiding inside.

Recipe Box

Simplifying Radicals Recipe

Step 1: Find the largest perfect square factor.

Step 2: Rewrite the number as

(perfect square)(leftover)

Step 3: Split the radical.

Step 4: Simplify the perfect square.

Perfect Squares To Know

See One

Simplify:

$\sqrt{72}$

Step 1

Find the largest perfect square factor.

72 = 36(2)

Step 2

Rewrite.

$\sqrt{72}$ = $\sqrt{36\cdot2}$

Step 3

Split the radical.

$\sqrt{72}$ = $\sqrt{36\cdot2}$ = $\sqrt{36}\space\times\sqrt2$

Step 4

Simplify.

$\sqrt{72}$ = $\sqrt{36\cdot2}$ = $\sqrt{36}\space\times\sqrt2$ = $6 \space\times\sqrt2$

Answer:

$\sqrt{72}=6\sqrt2$

Try One

Simplify:

45\sqrt{45}

45

Step 1

Largest perfect square factor?

45=9(5)45=9(5)

45=9(5)

Step 2

Rewrite.

45=95\sqrt{45} = \sqrt9\sqrt5

45

=9

5

Step 3

Simplify.

353\sqrt5

35

Answer:

45=35\sqrt{45}=3\sqrt5

45

=35

Do One

1

27\sqrt{27}

27

2

50\sqrt{50}

50

3

75\sqrt{75}

75

Pattern Watch

Notice what we're really doing.

72\sqrt{72}

72

↓

36⋅2\sqrt{36\cdot2}

36⋅2

↓

626\sqrt2

62

We're looking for a perfect square that can escape the radical.

Think:

Perfect squares can come out.

Leftovers stay inside.

Visual

Imagine a box.

┌───────────┐
│    72     │
└───────────┘

Look inside.

72 = 36 × 2

The 36 is a perfect square.

So it can leave.

6 √2

The 6 escapes.

The 2 stays trapped.

Mixed Check

1

Simplify:

18\sqrt{18}

18

A)

323\sqrt2

32

B)

232\sqrt3

23

C)

929\sqrt2

92

D)

626\sqrt2

62

2

Simplify:

32\sqrt{32}

32

A)

424\sqrt2

42

B)

828\sqrt2

82

C)

282\sqrt8

28

D)

16216\sqrt2

162

3

Simplify:

98\sqrt{98}

98

A)

727\sqrt2

72

B)

14214\sqrt2

142

C)

49249\sqrt2

492

D)

777\sqrt7

77

4

Which perfect square factor should be used first when simplifying

72\sqrt{72}

72

A)

44

4

B)

99

9

C)

3636

36

D)

6464

64

5

Simplify:

200\sqrt{200}

200

A)

10210\sqrt2

102

B)

20220\sqrt2

202

C)

1002100\sqrt2

1002

D)

10410\sqrt4

104

Tiny Reminder

When simplifying radicals:

Find

largest perfect square factor\text{largest perfect square factor}

largest perfect square factor

Rewrite

ab\sqrt{ab}

ab

Split

ab\sqrt a\sqrt b

a

b

Simplify

72=362=62\sqrt{72} = \sqrt{36}\sqrt2 = 6\sqrt2

72

=36

2

=62

Perfect squares come out. Leftovers stay in.

Future Connection

This exact skill appears later in:

• distance formula
• quadratic formula
• Pythagorean theorem
• circles
• SAT radicals
• SAT geometry
• Algebra 2 simplification

Students who can quickly spot:

72=36(2)72=36(2)

72=36(2)

or

98=49(2)98=49(2)

98=49(2)

will save enormous time later.