Laws of Exponents

Show your work for each problem.

1.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
Using the product rule: . When multiplying powers with the same base, we add the exponents.
2.
Simplify . Write your answer as where is the exponent.
Answer: 4
Using the quotient rule: . When dividing powers with the same base, we subtract the exponents.
3.
What is the value of ?
1. a)Undefined
2. b)
3. c)
4. d)
Answer:
. Any non-zero number raised to the power of 0 equals 1. This can be shown by: .
4.
Simplify using the power rule.
Answer: 729
1. What rule do we use for a power raised to a power? power rule
2. Multiply the exponents: 6
3. So 6
4. Calculate 729
5.
Write as a fraction.
Answer: 1/16
. A negative exponent means we take the reciprocal of the positive power.
6.
Which expression equals ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Using the quotient rule, we subtract the exponents when dividing.
7.
Simplify
Answer: 16
1. What rule applies when multiplying powers with the same base? product rule
2. Add the exponents: 2
3. So 2
4. Calculate 16
8.
Simplify . Calculate the final value.
Answer: 1000
. Or using the product to a power rule: .
9.
Simplify using the quotient rule.
Answer: 1
1. Apply the quotient rule: 5
2. What is ? 0
3. So . What is ? 1
10.
Simplify
1. a)
2. b)
3. c)
4. d)
Answer:
. The quotient to a power rule means we raise both numerator and denominator to the power.
11.
Simplify
Answer: a^6
1. First, simplify the numerator using the product rule. 8
2. Now we have . Apply the quotient rule: 2
3. What is ? 6
4. Final answer: 6
12.
Simplify . Give the numerical answer.
Answer: 16
. Then .
13.
Which expression is equivalent to ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Subtracting a negative is the same as adding!
14.
Simplify
Answer: 27
1. Apply the power rule to . What is ? 8
2. So 8
3. Now simplify using the quotient rule. What is ? 3
4. Calculate 27
15.
If and , what is ?
Answer: 256
, so . , so . Therefore . We can also use the product rule: .
16.
Evaluate
Answer: 100
1. What rule do we use when dividing powers with the same base? quotient rule
2. Apply the rule: 2
3. What is ? 2
4. Calculate 100