Zero Exponent

Show your work for each problem.

1.
What is $9^0$ ?
1. a)0
2. b)9
3. c)1
4. d)90
Answer: 1
$9^0 = 1$ . From $9^1 = 9$ , one step down divides by the base: $9 \div 9 = 1$ . It is not $0$ and not $9$ .
2.
Evaluate $25^0$ .
Answer: 1
$25 \neq 0$ , so $25^0 = 1$ .
3.
Which expression is equal to $1$ ?
1. a) $1^5 + 1$
2. b) $5^0$
3. c) $5^1$
4. d) $0 \times 5$
Answer: $5^0$
$5^0 = 1$ . The others are $5^1 = 5$ , $0 \times 5 = 0$ and $1^5 + 1 = 2$ .
4.
Evaluate $2^0 + 3^0$ .
Answer: 2
$2^0 + 3^0 = 1 + 1 = 2$ .
5.
You know that $4^3 = 64$ . Use the pattern to find $4^0$ .
Answer: 1
1. Divide $64$ by $4$ . What is $4^2$ ? 16
2. Divide by $4$ again. What is $4^1$ ? 4
3. One more step down. What is $4^0$ ? 1
6.
Evaluate $\frac{6^3}{6^3}$ with the quotient rule and check the result.
Answer: 1
1. Subtract the exponents. What is $3 - 3$ ? 0
2. So the quotient is $6^0$ . What is $6^0$ ? 1
3. Check: $6^3 = 216$ . What is $216 \div 216$ ? 1
7.
Simplify $5x^0$ (with $x \neq 0$ ).
1. a) $5x$
2. b) $1$
3. c) $5$
4. d) $0$
Answer: $5$
$5x^0 = 5 \times x^0 = 5 \times 1 = 5$ . Only $(5x)^0$ would be $1$ .
8.
What is $(-4)^0$ ?
1. a) $-4$
2. b) $0$
3. c) $-1$
4. d) $1$
Answer: $1$
The base is $-4$ , which is not zero, so $(-4)^0 = 1$ .
9.
Evaluate $-4^0$ (no brackets).
Answer: -1
$-4^0 = -(4^0) = -(1) = -1$ .
10.
Evaluate $(2x)^0 + 2x^0$ for $x = 5$ .
Answer: 3
1. The whole bracket is the base. What is $(2x)^0 = 10^0$ ? 1
2. Only $x$ has the exponent. What is $2x^0 = 2 \times 5^0$ ? 2
3. Add the two results. 3
11.
Evaluate $8^0 \times 8^2$ .
Answer: 64
1. What is $8^0$ ? 1
2. What is $8^2$ ? 64
3. Multiply the two values. 64
12.
Evaluate $-2^0 + (-2)^0$ .
Answer: 0
1. What is $-2^0$ (no brackets)? -1
2. What is $(-2)^0$ ? 1
3. Add the two results. 0
13.
A population of insects is modeled by $P = 300 \times 3^t$ , where $t$ is the time in weeks. How many insects are there at the start ( $t = 0$ )?
Answer: 300
1. What is $3^0$ ? 1
2. What is $300 \times 3^0$ ? 300
14.
Which statement is true? (Assume $y \neq 0$ .)
1. a) $6^0 = 0$
2. b) $(6y)^0 = 6$
3. c) $-6^0 = 1$
4. d) $6y^0 = 6$
Answer: $6y^0 = 6$
$6y^0 = 6 \times 1 = 6$ is true. The others are false: $-6^0 = -1$ , $(6y)^0 = 1$ and $6^0 = 1$ .
15.
Evaluate $\left(\frac{3}{4}\right)^0 \times 12 - 5^0$ .
Answer: 11
$1 \times 12 - 1 = 12 - 1 = 11$ .
16.
Find the exponent $n$ such that $7^n = 1$ .
Answer: 0
$7^0 = 1$ , so $n = 0$ . For $n = 1$ we get $7$ , and larger exponents give even larger values.
17.
Evaluate $\frac{3^4 \times 3^2}{3^6} + 5^0$ .
Answer: 2
1. Product rule: $3^4 \times 3^2 = 3^{\square}$ . What goes in the box? 6
2. Quotient rule: $\frac{3^6}{3^6} = 3^{\square}$ . What goes in the box? 0
3. What is $3^0$ ? 1
4. What is $1 + 5^0$ ? 2
18.
Evaluate $5x^0 + (5x)^0 - x^0$ for $x = 7$ .
Answer: 5
1. What is $5x^0$ ? 5
2. What is $(5x)^0$ ? 1
3. What is $x^0$ ? 1
4. Combine: $5 + 1 - 1$ . 5