Unit 7 Taking a Glance at Algebra

7.1 – Exponents & Roots

7.2 – Pythagorean Theorem

7.3 – Solving Multi-Step Equations

7.1 NOTES Exponents & Roots

First, let’s refresh ourselves on some definitions.

Base: _____________________________________ Exponent: __________________________________ Power: _____________________________________ And now some examples of the three forms of Exponents:

Exponential form: ___________________________ Expanded form: ____________________________ Standard form: _____________________________ Go ahead and work these examples: Example #1: Write the following in standard form: Example #2: Write the following in expanded form:

45 = ________ 73 = _________ Example #3: Write the following in exponential form: y • y • y • y = ____________

Exploration 2x 10x

24 = 104 = 23 = 103 = 22 = 102 = 21 = 101 = 20 = 100 = 2-1 = 10-1 = 2-2 = 10-2 =

Conclusions: ____________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ So, any base raised to the power of 0 (zero) = __________ And a base raised to a negative power is equal to the _____________________________ Other Exponent Rules Product of a Power When you multiply numbers with the SAME BASE, you can _____________________ Ex: 32 • 34 = _____________ = _____________ Power of a Power When you raise a power to a power, you can _________________________ Ex: (42)3 = _______________ = ______________

When you raise a product to a power, you can _______________________

Ex: (x • y)3 = _____________ When you divide numbers with the same base, you can _____________________

Ex: 67 ÷ 65 = ___________ = ___________

Raising a Quotient to a Power

Ex:

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xy⎛

⎝ ⎜ ⎞

⎠ ⎟

3

= _______________ = _____________

Guided Practice Evaluate each of the following Exponent problems: 1) x-4 = ____________ 2) 5-3 = ____________ 3) y3 • y4 = __________ 4) 22 • 23 = ____________ 5) (w3)4 = ____________ 6) (34)2 = _____________

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a8

a5 = ___________ 8)

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49

47 = _____________

9) n4 • n-6 = __________ 10) 570 = ____________

Joey says that

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5−4 is -625.

Anna says that

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5−4 is equivalent to the fraction

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1625 .

Alex says that

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5−4 is equivalent to the fraction

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−1625 .

Who do you think is correct? Why? _______________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Roots

A square has an area of 9 cm2. What is the length of each side of the square? A square has an area of 16 cm2. What is the length of each side of the square? How can this help us with square roots? Let’s fill-in the table to the right. When we work backwards, we can use the table to see that

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49 is 7, and that

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64 is 8. But what about all of the square roots in between 49 & 64? Well, we can estimate those based on the roots that we do know. So we can say that

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55 is somewhere in between 7 and 8, and probably a little closer to 7. Write your estimate here __________ What about the square root of 72? ________________________

Guided Practice

Evaluate each of the following:

1)

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196 = ___________

2)

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30 = approximately ____________

3)

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289 = ___________

4)

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529 = ___________ 5)

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112 is between ______ and _______

7.2 NOTES Pythagorean Theorem

a2 + b2 = c2

This formula is called the Pythagorean Theorem. But what does it mean, and how can we use it? Let’s look at how right triangles are set-up first, then we will answer that question.

In every right triangle, the longest side (the side opposite the right angle) is called the HYPOTENUSE. The other two sides are each called a LEG. Label the sides of the right triangle shown below: The Pythagorean Theorem describes the relationship of the lengths of sides of a right triangle, and can be used to find the missing length in any right triangle.

Example 1: Use the Pythagorean Theorem to find the missing side length: Example 2: Use the Pythagorean Theorem to find the missing side length:

Guided Practice

Find the missing side of each of the following right triangles, using the Pythagorean Theorem:

1) 2) 3) 4)

7.3 NOTES Solving Multi-Step Equations

Let’s start by reviewing something we worked on way back in Unit 2, Solving Equations! How would you solve the equation below?

3y + 7 = 40

y = ____________ Great job! Now what if the equation looked like this?

6(7j + 3) = -28 + 19j Here are some helpful hints, from your soon-to-be 8th grade math teachers:

Steps to Solving Multi-Step Equations 1. Always distribute into parentheses first if you can

2. Combine your like terms on each side of the equals sign

3. Move your terms with variables to one side of the equals sign to work towards a two-step equation

4. Undo addition/subtraction by using inverse operations

5. Undo Multiplication/Division by using inverse operations

Don’t forget the golden rule to solving equations: Whatever we do to one side of the equals sign, we have to do exactly the same to the other side! Ok, so using these steps, let’s solve the problem from above:

6(7j + 3) = -28 + 19j

j = _______

Guided Practice Solve the following multi-step equations: 1) 19 – h – h = -13 2) 5n – 16 – 8n = -10 3) 14 + 6a – 8 = 18 4) 6(3m + 5) = 66 5) 3(4y – 8) = 12 6) 17 – 9y = -3 + 16y

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Name: _______________________________________________ Date: __________ Hr: ________

7.1 Homework 1. Simplify each of the following: a) x-2 = ___ b) y-4 = ___ c) w0 = ___

2. Simplify each of the following: a) 3-2 = ______ b) 4-3 = ______ c) 570 = ______

3. Simplify each of the following expressions: a) x2 • x3 = _________ b) (y4)3 = __________ c) z6 ÷ z4 = _________

4. Simplify each of the following expressions: a) 33 • 32 = _____________ b) (22)3 = ____________ c) 86 ÷ 84 = ____________

5. Simplify each of the following expressions: a) 23 • 2-5 = ____________ b) 44 ÷ 47 = ____________

6. Find each of the following square roots (round to the nearest tenths place, if necessary). a) 25 = ________ b) 361 = _______ c) 114 = _______

Name: _______________________________________________ Date: __________ Hr: ________

7.2 Homework 1. Use the Pythagorean Theorem to find the missing side of the triangle shown below:

2. Use the Pythagorean Theorem to find the missing side of the triangle shown below:

3. Use the Pythagorean Theorem to find the missing side of the triangle shown below:

4. Use the Pythagorean Theorem to find the missing side of the triangle shown below:

5. Use the Pythagorean Theorem to find the missing side of the triangle shown below:

6. Josh lives at the house marked with an X on the map shown below. He wants to get some frozen yogurt. How much shorter would it be to cut across the park, then to walk down each of the streets?

Name: _______________________________________________ Date: __________ Hr: ________

7.3 Homework 1. Solve the following equation:

9h – 6 + 2h = 27

2. Solve the following equation:

-5x + 2x + 3 = 15

3. Solve the following equation:

-7(3b + 2) = 49

4. Solve the following equation:

-2(3x – 4) + 3 = 5

5. Solve the following equation:

5x – 3 = 2x + 12

6. Solve the following equation:

3(4x – 2) = 18 – 12x

7. Solve the following equation:

6x + 14 = 8x – 2x – 12

8. The sum of three consecutive integers is 357. What are the three integers? Write an equation that illustrates the situation and solve it.

Name: ________________________________ Hour: ______________

Zero as an Exponent

Negative Exponent

Multiplying Powers with the SAME BASE

Raising a Power to a Power

Raising a Product to a Power

Dividing Powers with the SAME BASE

Raising a Quotient to a Power