NAME _____________________________________________ DATE ____________________________ PERIOD _____________
Review Math 7/8
Multiply and Divide
The Product of Powers rule states that to multiply powers with the same base, add their exponents.
Example 1
Simplify. Express using exponents.
a. 𝟐𝟑 • 𝟐𝟐
23 • 22 = 23+2
The common base is 2.
= 25
Add the exponents.
The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents.
Example 2
Simplify
𝒌𝟖
.
𝒌
Express using exponents.
𝑘8
𝑘1
= 𝑘 8−1 The common base is k.
= 𝑘7
Subtract the exponents.
Powers of Powers
Power of a Power: To find the power of a power, multiply the exponents.
Power of a Product: To find the power of a product, find the power of each factor and multiply.
Example 1
Simplify (𝟓𝟑 )𝟔 .
(53 )6 = 53 • 6
Power of a power
= 518
Simplify.
Simplify. Express using exponents.
1. 52 • 56
5.
25 • 37 • 43
21 • 35 • 4
3
9. (5 𝑎6 𝑏 9 )2
2. 𝑒 2 • 𝑒 7
415 • (−5)6
3. 2𝑎5 • 6𝑎
67 • 76 • 85
4. 4𝑥 2 • (−5𝑥 6 )
(−3)6 • 105
6. 412 • (−5)4
7. 65 • 75 • 84
8. (−3)4 • 103
10. (4𝑥 2 )3 (3𝑥 6 )4
11. (0.6𝑝5 )3
12. (5 𝑤 5 𝑥 3 )2
1
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
Negative Exponents
Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the multiplicative inverse of
the number to the nth power.
Example 1
Write each expression using a positive exponent.
a. 𝟕−𝟑
1
7−3 = 73
Definition of negative exponent
b. 𝒂−𝟒
1
𝑎−4 = 𝑎3
Definition of negative exponent
Simplify. Express using positive exponents.
13.
65
62
14. 𝑛−2 • 𝑛−3
15.
𝑤3
𝑤 −1
16.
𝑘 −4
𝑘 −6
Roots
A square root of a number is one of its two equal factors. A radical sign, √ is used to indicate a positive square root.
Every positive number has both a negative and positive square root.
Examples
Find each square root.
1. √1
Find the positive square root of 1; 12 = 1, so √1 = 1.
2. −√16
Find the negative square root of 16; (–4)2 = 16, so − √16 = −4.
3. ±√0.25
Find both square roots of 0.25; 0.52 = 0.25, so ±√0.25 = ±0.5.
4. √−49
There is no real square root because no number times itself is equal to –49.
Find each square root.
17. √4
18. √9
ALGEBRA Solve each equation. Check your solution(s).
19. 𝑥 2 = 121
20. 𝑎2 = 3,600
The Pythagorean Theorem
The Pythagorean Theorem describes the relationship between the lengths of the legs and the hypotenuse for any
right triangle. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the
lengths of the legs. You can use the Pythagorean Theorem to find the length of a side of a right triangle if the
lengths of the other two sides are known.
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
Write an equation you could use to find the length of the missing side of each right triangle. Then find
the missing length. Round to the nearest tenth if necessary.
1.
2.
a2 + b2 = c2
242 + 322 = c2
576 + 1,024 = c2
1,600 = c2
± √1,600 = c
c = 40 or –40
a2 + b2 = c2
152 + b2 = 202
225 + b2 = 400
225 + b2 – 225 = 400 – 225
b2 = 175
√𝑏 2 = ±√175
b ≈ ± 13.2
Write an equation you could use to find the length of the missing side of each right triangle. Then find
the missing length. Round to the nearest tenth if necessary.
1.
2.
3.
Lines
• Perpendicular lines are lines that intersect at right angles.
• Parallel lines are two lines in a plane that never intersect or cross.
• A line that intersects two or more other lines is called a transversal.
• If the two lines cut by a transversal are parallel, then these special pairs of angles are congruent:
alternate interior angles, alternate exterior angles, and corresponding angles.
Example 1
Classify ∠4 and ∠8 as alternate interior, alternate exterior, or corresponding.
∠4 and ∠8 are in the same position in relation to the
transversal on the two lines. They are corresponding angles.
Exercises
In the figure at the right, line m and line n are parallel.
If m∠3 = 64°, find each given angle measure.
Justify each answer.
1. m∠8
3. m∠4
2. m∠10
4. m∠6