Pre Algebra Review

Pythagorean Theorem & It's Converse

October 11th, 2021

The Pythagorean Theorem

Hypotenuse

• The longest side of a right triangle

• The side that is opposite (across) from the 90^o angle.

• When using the Pythagorean Theorem, the Hypotenuse is represented by the variable, c.

Leg (of a Triangle)

• The two shorter sides of a right triangle

• The sides that are adjacent to the 90^o angle.

• When using the Pythagorean Theorem, the Legs are represented by the variables, a & b.

Is it a Right Triangle?

Multiple Choice

Is this a right triangle?

• Yes

• No

Multiple Choice

Is this a right triangle?

• Yes

• No

Multiple Choice

Is a triangle with side lengths of 12, 9, and 15 a right triangle?

• Yes

• No

Multiple Choice

Is a triangle with side lengths of 7, 4, and 4 a right triangle?

• Yes

• No

Multiple Choice

The Pythagorean Theorem ONLY works on which triangle? 

• obtuse

• scalene

• isosceles

• right

Multiple Choice

Side c on a right triangle is ALWAYS the longest.

• true

• false

Multiple Choice

What is the relationship between the sides in the Pythagorean Theorem shown?

Hint: Think about when the hypotenuse is missing!

• a + b = c

• a² + b² = c²

• a² – b² = c²

• a – b = c

Multiple Choice

Which equation will help you find the missing side?

• $$4^2+10^2=c^2$$

• $$10^2-4^2=c^2$$

• $$c^2=100+10^2$$

• $$c^2+4^2=10^2$$

Multiple Choice

Please read the question carefully:

• A

• B

• C

• D

Multiple Choice

Use the Pythagorean Theorem to solve for x.

• $$\sqrt{1213}=34.8$$

• $$\sqrt{455}=21.3$$

• $$\sqrt{87}=9.3$$

• $$\sqrt{49}=7$$

Multiple Choice

Which equation is set up correctly for this word problem?

• $$23^2+b^2=10^2$$

• $$10^2+b^2+23^2$$

• $$10^2+23^2=c^2$$

• $$a^2+23^2=c^2$$

Multiple Choice

Which equation will help you find the missing side?

• $$4^2+10^2=c^2$$

• $$10^2-4^2=c^2$$

• $$c^2=100+10^2$$

• $$c^2+4^2=10^2$$

Multiple Choice

Which of these is closest to the length of PQ?

• 10.9 units

• 13 units

• 2.4 units

• 169 units

Multiple Choice

A rectangular pool at Andy's gym measures 25 m by 50 m. How far would Andy swim if he swims across the pool diagonally?

• 5 m

• 8.7 m

• 43.3 m

• 55.9 m

Multiple Choice

 Find the distance between these 2 points.

• 5.8

•  6.3

• 8.1

• 40

Multiple Choice

_{^{Find the missing side}}

• 9

• 41

• 6.4

• 3

Multiple Choice

• 7 ft

• 5 ft

• 3 ft

• 2 ft

Multiple Choice

What is the height of the cone? 

• 14 cm

• 8 cm

• 12 cm

• 13.9 cm

Multiple Choice

_{^{Find the missing side}}

• 48

• 108

• 83.6

• 72

Multiple Choice

Casey travels from her house directly west to the bank and then directly north from the bank to the mall. She then travels home on the road connecting the mall and her house. What is her total distance traveled?

• 17 miles

• 32 miles

• 37 miles

• 40 miles

Multiple Choice

• 9.2 in

• 11 in

• 13 in

• 9.6 in

Multiple Choice

The Pythagorean Theorem is
a² + b³ = c⁴

• false

• true

Product & Quotient Rules of Exponents

Laws of Exponents

Product Rule

• Look at the right and see what happens when you multiply powers that have the same base

• When you expand 3² you have two 3s being multiplied and when you expand 3³ you have three 3s being multiplied.

• All together how many 3s are you multiplying?

Product Rule Short Cut?!!

When you multiply same base powers you keep the base and add the exponents.

x² * x⁷ = x²⁺⁷ = x⁹

Multiple Choice

Simplify the following:

$$a^{11}\cdot a^2\cdot a^0$$  

• $$a^{13}$$  

• $$1$$  

• $$a^{22}$$  

• $$a^0$$  

Multiple Choice

Simplify the following:

$$x^3\cdot y^2\cdot x^2$$  

• $$xy^7$$  

• $$x^5y^2$$  

• $$\left(xy\right)^7$$  

• $$x^5$$  

Multiple Choice

Simplify the following:

$$x^3\cdot x\cdot x^{-2}$$  

• $$x^1$$  

• $$x^5$$  

• $$x^6$$  

• $$x^2$$  

Quotient Rule

• When dividing powers that have the same base think of dividing as the inverse of multiplying, so what is the inverse of adding?

• When you expand x⁵ you have five Xs being multiplied in the numerator and when you expand x² you have two Xs being multiplied in the denominator.

• So what happens when you have the Xs in the numerator and Xs in the denominator? Yup! you cross out the matching pairs!

• Now what did you end up with?

Quotient Rule Short Cut?!!

When you divide same base powers you keep the base and subtract the exponents.

x⁷ / x² = x^7-2 = x⁵

Multiple Choice

Simplify the following:

$$\frac{x^{16}}{x^2}$$  

• $$x^{14}$$  

• $$x^8$$  

• $$x^{18}$$  

• $$x^{-8}$$  

Multiple Choice

Simplify the following:

$$\frac{x^9y^7}{x^2y^6}$$  

• $$xy^8$$  

• $$\left(xy\right)^{24}$$  

• $$x^7y$$  

• $$x^{11}y^1$$  

Multiple Choice

Simplify the following:

$$\frac{x^7}{x^{-4}}$$  

• $$x^{11}$$  

• $$x^3$$  

• $$x^{-3}$$  

• $$x^{-11}$$  

Multiple Choice

Simplify the following:

$$\frac{x^3}{x^8}$$  

• $$x^{-5}$$  

• $$x^5$$  

• $$x^{-3}$$  

• $$\frac{1}{x^{11}}$$  

Multiple Choice

$$365^0=$$  

• 0

• 1

• 365

• $$\frac{1}{365}$$  

Multiple Choice

$$x^{-3}\cdot x^7\cdot x^2$$  =

• $$x^6$$  

• $$x^{12}$$  

• $$x^{-42}$$  

• $$x^{-12}$$  

Multiple Select

According to the exponent rules, when we raise a Power to a Power(Power Rule) we ______ the exponents.

• Add

• Subtract

• Multiply

• Divide

Multiple Select

According to the product rule, when we multiply the number with exponents with the same base we _______ the exponents.

• Add

• Subtract

• Multiply

• Divide

Multiple Select

The quotient law states: Keep the base the same and _________ the exponents.

• Add

• Multiply

• Subtract

• Divide

Multiple Choice

Anything raised to a power of zero is always:

• 0

• itself

• 1

• negative

Multiple Choice

Simplify the expression.

$$x^4\left(x^{12}\right)$$  

• $$x^3$$  

• $$x^{48}$$  

• $$x^{16}$$  

• $$x^8$$  

Multiple Choice

 Simplify 

$$\left(-4r^5\right)\left(2r^8\right)$$  

• $$-2r^{13}$$  

• $$-8r^{40}$$  

• $$-2r^{40}$$  

• $$-8r^{13}$$  

Multiple Choice

Simplify: 

$$\frac{k^4}{k^3}$$  

• $$k$$  

• $$k^7$$  

• $$k^{12}$$  

• $$\frac{1}{k}$$  

Multiple Choice

Simplify:

$$\frac{10h^9}{5h^7}$$  

• $$\frac{2}{h^2}$$  

• $$15h^{16}$$  

• $$2h^2$$  

• undefined 

Multiple Choice

Simplify:

$$\frac{4x^{12}y^3z^6}{2x^4y^3z}$$  

• $$2x^8z^5$$  

• $$\frac{2y^6}{x^3z^5}$$  

• $$2x^3y^6z^5$$  

• $$\frac{z^5}{2x^8}$$  

Multiple Choice

Simplify:

$$\left(x^3\right)^8$$  

• $$x^{11}$$  

• $$x^{\frac{8}{3}}$$  

• $$x^{24}$$  

• $$x^5$$  

Multiple Choice

Simplify:

$$\left(3xy^3\right)^2$$  

• $$3x^2y^5$$  

• $$9x^2y^6$$  

• $$6x^2y^6$$  

• $$-9x^2y^5$$  

Multiple Choice

Simplify:

$$\frac{6x^{-2}y^3}{2x^5y^{-1}}$$  

• $$\frac{3y^4}{x^7}$$  

• $$\frac{4y^4}{x^7}$$  

• $$8x^7y^4$$  

• $$\frac{4y^4}{x^7}$$  

Multiple Choice

$$\left(-4x\right)^0$$  

• -1

• 1

• 0

• -4

Multiple Choice

$$4a^3\ \cdot\ 2a^4$$  

• $$8a^7$$  

• $$8a^{12}$$  

• $$6a^7$$  

• $$6a^{12}$$  

Multiple Choice

$$\frac{10z^5}{5z^2}$$  

• $$5z^3$$  

• $$2z^3$$  

• $$5z^7$$  

• $$15z^7$$  

Multiple Choice

$$\left(x^5y^6\right)^2$$  

• $$x^7y^8$$  

• $$x^{10}y^{12}$$  

• $$x^{25}y^{36}$$  

• $$x^3y^4$$