Product Rule for Exponents

Show your work for each problem.

1.
Which expression is equal to $5^2 \cdot 5^6$ ?
1. a) $5^8$
2. b) $10^8$
3. c) $5^{12}$
4. d) $25^8$
Answer: $5^8$
Same base $5$ , so add the exponents: $5^{2+6} = 5^8$ . $5^{12}$ multiplies the exponents, and $25^8$ and $10^8$ change the base.
2.
$2^3 \cdot 2^4 = 2^{\square}$ . What number goes in the box?
Answer: 7
$2^3 \cdot 2^4 = 2^{3+4} = 2^7$ . Check: $8 \cdot 16 = 128 = 2^7$ .
3.
Which product can be written as a single power with the product rule?
1. a) $2^3 \cdot 5^2$
2. b) $5^2 \cdot 3^3$
3. c) $7^3 \cdot 7^5$
4. d) $2^4 + 2^3$
Answer: $7^3 \cdot 7^5$
Only $7^3 \cdot 7^5$ is a product with the same base: $7^{3+5} = 7^8$ . $2^3 \cdot 5^2$ and $5^2 \cdot 3^3$ have different bases, and $2^4 + 2^3$ is a sum, not a product.
4.
$x \cdot x^6 = x^{\square}$ . What is the exponent?
Answer: 7
$x = x^1$ , so $x^1 \cdot x^6 = x^{1+6} = x^7$ .
5.
Write $3^2 \cdot 3^3$ as a single power and find its value.
Answer: 243
1. How many factors of 3 are in $3^2$ ? 2
2. How many factors of 3 are in $3^3$ ? 3
3. Together that is $2 + 3$ factors. Complete $3^2 \cdot 3^3 = 3^{\square}$ . 5
4. What is the value of $3^5$ ? 243
6.
Simplify $10^2 \cdot 10^4$ and write the result as an ordinary number.
Answer: 1000000
1. Both powers have base 10. Add the exponents $2 + 4$ . 6
2. Write $10^6$ as an ordinary number. 1000000
7.
Simplify $4x^3 \cdot 2x^5$ .
1. a) $8x^{15}$
2. b) $8x^8$
3. c) $6x^8$
4. d) $8x^2$
Answer: $8x^8$
Multiply the coefficients: $4 \cdot 2 = 8$ . Add the exponents: $x^{3+5} = x^8$ . So $4x^3 \cdot 2x^5 = 8x^8$ .
8.
$y^4 \cdot y^2 \cdot y^3 = y^{\square}$ . What is the exponent?
Answer: 9
$4 + 2 + 3 = 9$ , so the product is $y^9$ .
9.
Sam writes $2^3 \cdot 2^4 = 4^7$ . What went wrong?
1. a)He should have multiplied the exponents
2. b)He should have subtracted the exponents
3. c)Nothing, the answer is correct
4. d)He multiplied the bases instead of keeping the base 2
Answer: He multiplied the bases instead of keeping the base 2
The base must stay 2: $2^3 \cdot 2^4 = 2^7 = 128$ . Sam's answer $4^7 = 16\,384$ is far too big. Adding the exponents, $3 + 4 = 7$ , was right; only the base was changed.
10.
Simplify $3x^2 \cdot 4x^3$ , then find its value for $x = 2$ .
Answer: 384
1. Multiply the coefficients $3 \cdot 4$ . 12
2. Add the exponents of $x$ : $2 + 3$ . 5
3. So the product is $12x^5$ . What is its value for $x = 2$ ? 384
11.
Simplify $5a \cdot 2a^4$ .
Answer: 10a^5
1. Multiply the coefficients $5 \cdot 2$ . 10
2. Which exponent does $a$ have in $5a$ ? 1
3. Add the exponents of $a$ : $1 + 4$ . 5
12.
A rectangle is $2^5$ cm long and $2^3$ cm wide. Find its area in cm².
Answer: 256
1. The area is $2^5 \cdot 2^3$ . Write it as $2^{\square}$ : what is the exponent? 8
2. What is $2^8$ ? 256
13.
$3^{\square} \cdot 3^4 = 3^{10}$ . What number goes in the box?
Answer: 6
The exponents add up: $\square + 4 = 10$ , so the missing exponent is $6$ . Check: $3^6 \cdot 3^4 = 3^{6+4} = 3^{10}$ .
14.
Which product is NOT equal to $2^{10}$ ?
1. a) $2 \cdot 2^9$
2. b) $2^4 \cdot 2^6$
3. c) $2^2 \cdot 2^5$
4. d) $2^5 \cdot 2^5$
Answer: $2^2 \cdot 2^5$
$2^2 \cdot 2^5 = 2^{2+5} = 2^7 = 128$ , not $2^{10} = 1024$ . Getting $10$ here would mean multiplying the exponents $2 \cdot 5$ , which is the classic mistake. The others work: $4 + 6 = 10$ , $1 + 9 = 10$ and $5 + 5 = 10$ .
15.
Find the value of $(-2)^3 \cdot (-2)^4$ .
Answer: -128
$(-2)^3 \cdot (-2)^4 = (-2)^{3+4} = (-2)^7$ . Seven negative factors give a negative result: $(-2)^7 = -128$ . Check: $-8 \cdot 16 = -128$ .
16.
Simplify $2a^3b \cdot 5a^4b^2$ .
Answer: 10a^7b^3
1. Multiply the coefficients $2 \cdot 5$ . 10
2. Exponent of $a$ : add $3 + 4$ . 7
3. Exponent of $b$ : add the exponents of $b$ and $b^2$ . 3
17.
Write $4^3 \cdot 2^5$ as a single power of 2 and find its value.
Answer: 2048
1. Write 4 as a power of 2: $4 = 2^{\square}$ . What is the exponent? 2
2. $4^3 = 4 \cdot 4 \cdot 4 = 2^2 \cdot 2^2 \cdot 2^2$ . Write it as $2^{\square}$ : what is the exponent? 6
3. Now $2^6 \cdot 2^5 = 2^{\square}$ . What is the exponent? 11
4. What is $2^{11}$ ? 2048
18.
Find $n$ if $x^n \cdot x^3 \cdot x^n = x^{11}$ .
Answer: 4
1. The exponents add up: $n + 3 + n = 11$ . What is $2n$ ? 8
2. What is $n$ ? 4