9.0 Geometry

Perfect Squares and Square Roots

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Perfect Squares

• Numbers that result from multiplying a whole number by itself.

• 16 is a PERFECT SQUARE because 4² = 16

• 121 is a PERFECT SQUARE because 11² = 121

Practice!

On the following slides, find the perfect squares.

Multiple Choice

Find 5²

• 15

• 25

• 20

• 10

Multiple Choice

Find 10²

• 100

• 20

• 200

• 10

Multiple Choice

Find 9²

• 18

• 81

• 90

• 64

Multiple Choice

Find 8²

• 68

• 64

• 16

• 56

Square Roots

Practice!

On the following slides, find the square roots.

Multiple Choice

Ex 1- $$\sqrt{49}$$  

• 7

• 3

• 4

• 6

Multiple Choice

Ex 2- $$\sqrt{400}$$  

• 20

• 100

• 4

• 200

Multiple Choice

Ex 3- $$\sqrt{36}$$  

• 3

• 6

• 7

• 18

Multiple Choice

Ex 4- $$\sqrt{144}$$  

• 12

• 100

• 11

• 13

Poll

How do you feel about finding perfect squares and square roots?

Rational Exponents and Simplifying Radicals

Rational Exponents

• The numerator (top of the fraction) is the exponent

• Hint: exponent goes at the TOP of the number

• The denominator (bottom of the fraction) is the root

• Hint: the roots are at the BOTTOM of the tree.

Multiple Choice

Write the expression in radical form.

x^3/4

• ∜x³

• ∛x⁴

• x^.75

• √x²

Multiple Choice

Write in exponential form

∛x⁷

• $$x^{\frac{7}{3}}$$

• $$x^{\frac{3}{7}}$$

• 3x⁷

• 7x³

Multiple Choice

• 7³

• 7

• 7^3/4

• 7^4/3

Multiple Choice

HINT: what exponent is at the 2? It is "invisible"

• 2^1/6

• 64

• 6^1/2

• 2⁶

Multiple Choice

HINT: what root is it? It is "invisible"

• 216

• 6^1/2

• 6²

• 6^3/2

Multiple Choice

HINT: both are invisible.

• 7²

• 7^1/2

• 7^-2

• 7^-1/2

​Simplifying Radicals

1. ​Radicals, in our case, square roots must be simplified as much as possible.

2. ​Simplifying radicals removes as many perfect squares or "pairs" as possible.

3. ​To simplify, break the radicand into factors.

4. ​Continue to factor until only prime numbers or perfect squares remain

5. ​Place "pairs" or perfect squares on the "outside"

6. ​Remember to multiply all coefficients together!

Multiple Select

What is the prime factorization of

$$\sqrt{56}$$ ?

• 2*4*7

• 2*2*2*7

• 4*7

Multiple Choice

Simplify $$\sqrt{56}$$  

• $$2\sqrt{14}$$  

• $$4\sqrt{14}$$  

• $$2\sqrt{28}$$  

• $$2\sqrt{9}$$  

Multiple Choice

Simplify $$\sqrt[]{32}$$  

• $$\sqrt[]{32}$$  

• $$16\sqrt[]{2}$$  

• $$4\sqrt[]{2}$$  

• $$2\sqrt[]{8}$$  

Multiple Choice

Simplify $$\sqrt[]{50}$$  

• $$5\sqrt[]{2}$$  

• $$25\sqrt[]{2}$$  

• $$\sqrt[]{5\cdot10}$$  

• $$\sqrt[]{50}$$  

Multiple Choice

Simplify $$\sqrt[]{60}$$  

• $$2\sqrt[]{15}$$  

• $$2\sqrt[]{30}$$  

• $$30\sqrt[]{2}$$  

• $$\sqrt[]{6\cdot10}$$  

Poll

How comfortable are you with simplifying radicals?

• I understand fully

• I am okay but will get it with more practice

• I can do it but I am not fast

• I am lost