Zero Exponent

The **zero exponent rule** says that any nonzero number raised to the power of $0$ equals $1$: $$a^0 = 1 \quad (a \neq 0)$$ So $7^0 = 1$, $100^0 = 1$, $(-5)^0 = 1$ and $\left(\frac{2}{3}\right)^0 = 1$. **Why? Look at the pattern.** Every time the exponent goes down by $1$, you divide by the base: | Power | Value | |---|---| | $2^4$ | $16$ | | $2^3$ | $8$ | | $2^2$ | $4$ | | $2^1$ | $2$ | | $2^0$ | $2 \div 2 = 1$ | **The quotient rule agrees.** Dividing a power by itself gives $1$, and the quotient rule gives exponent $0$: $$\frac{a^n}{a^n} = a^{n-n} = a^0 \quad \text{and} \quad \frac{a^n}{a^n} = 1$$ **Watch what the exponent belongs to.** An exponent only applies to the base directly in front of it: - $5x^0 = 5 \times x^0 = 5 \times 1 = 5$, but $(5x)^0 = 1$ - $-3^0 = -(3^0) = -1$, but $(-3)^0 = 1$ **Special case:** $0^0$ is left undefined at this level. The rule $a^0 = 1$ is only for $a \neq 0$.

Follow the Pattern Down to $3^0$

Use the pattern of powers of $3$ to find $3^0$.

Start with a power you know: $3^3 = 3 \times 3 \times 3$ = $3^3 = 27$

Lower the exponent by 1: divide by 3: $27 \div 3 = 9$ = $3^2 = 9$

Lower it again: divide by 3: $9 \div 3 = 3$ = $3^1 = 3$

One more step down to exponent 0: $3 \div 3 = 1$ = $3^0 = 1$

Answer: $3^0 = 1$

Using the Quotient Rule

Evaluate $\frac{7^4}{7^4}$ in two ways.

Any nonzero number divided by itself: $\frac{7^4}{7^4} = \frac{2401}{2401}$ = $1$

Apply the quotient rule (same base): $\frac{7^4}{7^4} = 7^{4-4}$ = $7^0$

Both answers must be equal: $7^0 = \frac{7^4}{7^4} = 1$ = $7^0 = 1$

Answer: $\frac{7^4}{7^4} = 7^0 = 1$

$5x^0$ Versus $(5x)^0$

Evaluate $5x^0$ and $(5x)^0$ for $x = 2$.

In $5x^0$, the exponent belongs only to $x$: $5x^0 = 5 \times x^0 = 5 \times 2^0$ = $5 \times 1$

Multiply: $5 \times 1 = 5$ = $5x^0 = 5$

In $(5x)^0$, the exponent belongs to the whole bracket: $(5 \times 2)^0 = 10^0$ = $(5x)^0 = 1$

Answer: $5x^0 = 5$ and $(5x)^0 = 1$

$-3^0$ Versus $(-3)^0$

Evaluate $-3^0$ and $(-3)^0$.

Without brackets, the exponent belongs to $3$ only: $-3^0 = -(3^0)$ = $-(1)$

Apply the minus sign: $-(1) = -1$ = $-3^0 = -1$

With brackets, the base is $-3$: $(-3)^0$ has a nonzero base = $(-3)^0 = 1$

Answer: $-3^0 = -1$ and $(-3)^0 = 1$

An Expression With Several Zero Exponents

Evaluate $4^0 + 2^3 \times 6^0 - \left(\frac{1}{2}\right)^0$.

Replace every zero power by 1: $4^0 = 1$, $6^0 = 1$, $\left(\frac{1}{2}\right)^0 = 1$ = $1 + 2^3 \times 1 - 1$

Evaluate the remaining power: $2^3 = 8$ = $1 + 8 \times 1 - 1$

Multiply first: $8 \times 1 = 8$ = $1 + 8 - 1$

Add and subtract from left to right: $1 + 8 - 1$ = $8$

Answer: $8$

The zero exponent turns up wherever powers are used: - **Place value**: the ones place is $10^0 = 1$, so $345 = 3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0$ - **Computers**: the rightmost digit of a binary number is worth $2^0 = 1$ - **Growth models**: in $P = 500 \times 2^t$, the starting value at $t = 0$ is $500 \times 2^0 = 500$ - **Algebra**: simplifying $\frac{x^4}{x^4}$ or $3x^0$ needs the rule every time If $a^0$ were $0$, the ones place would be worth nothing and every growth model would start at zero. The value $1$ is the only one that keeps all the exponent rules working.

Mistake: Writing $a^0 = 0$, for example $9^0 = 0$

Why: The exponent $0$ does not mean multiplying by zero. Going down the pattern means dividing by the base, and $9 \div 9 = 1$.

Correct: $9^0 = 1$, because $\frac{9^1}{9^1} = 9^0$ and $\frac{9}{9} = 1$

Mistake: Writing $a^0 = a$, for example $9^0 = 9$

Why: That is the rule for exponent $1$: $9^1 = 9$. Exponent $0$ is one more step down, so you divide by $9$ once more.

Correct: $9^1 = 9$ but $9^0 = 1$

Mistake: Writing $5x^0 = 1$

Why: The exponent belongs only to $x$, not to the coefficient $5$. Only $x^0$ becomes $1$.

Correct: $5x^0 = 5 \times 1 = 5$, while $(5x)^0 = 1$

Mistake: Writing $-3^0 = 1$

Why: Without brackets, the base is $3$, not $-3$. The minus sign stays in front of the result.

Correct: $-3^0 = -(3^0) = -1$, while $(-3)^0 = 1$

Place Value and Powers of Ten

Every place in our number system is a power of ten. The ones place is $10^0 = 1$.

$6072 = 6 \times 10^3 + 0 \times 10^2 + 7 \times 10^1 + 2 \times 10^0$, and the last term is $2 \times 1 = 2$.

Binary Numbers in Computers

Computers store numbers in binary. Each binary place is a power of two, and the rightmost place is $2^0 = 1$.

Binary $1011$ means $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11$.

Starting Value of a Growth Model

Biologists model a population that doubles every hour with $P = P_0 \times 2^t$, where $t$ is the time in hours.

For $P = 200 \times 2^t$, after 3 hours $P = 200 \times 2^3 = 1600$, and at the start $P = 200 \times 2^0 = 200$.

For any nonzero number $a$: $a^0 = 1$

Pattern: each step down in the exponent divides by the base, so $2^1 = 2$ becomes $2^0 = 2 \div 2 = 1$

Quotient rule: $\frac{a^n}{a^n} = a^{n-n} = a^0$, and a number divided by itself is $1$

The exponent belongs only to the base right in front of it: $5x^0 = 5$ but $(5x)^0 = 1$

Brackets matter with negatives: $-3^0 = -1$ but $(-3)^0 = 1$

$a^0$ is never $0$ and never $a$; $0^0$ is left undefined at this level

Q: Why is $a^0$ equal to 1 and not 0?

A: Because each step down in the exponent divides by the base. From $5^1 = 5$, one more step down gives $5 \div 5 = 1$. The quotient rule says the same thing: $\frac{5^3}{5^3} = 5^0$, and any nonzero number divided by itself is $1$.

Q: What is $0^0$?

A: At this level $0^0$ is left undefined. The pattern argument needs us to divide by the base, and you cannot divide by $0$. That is why the rule is written $a^0 = 1$ for $a \neq 0$.

Q: Does the rule work for negative numbers and fractions?

A: Yes, as long as the base is not zero and the whole base is in brackets: $(-8)^0 = 1$ and $\left(\frac{3}{4}\right)^0 = 1$.

Q: Why does my calculator give $-1$ for $-3^0$?

A: The calculator follows the order of operations: the power comes before the minus sign, so it computes $-(3^0) = -1$. Type $(-3)^0$ with brackets to get $1$.

Exit Presentation

Zero Exponent

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Arithmetic / Exponents & Roots

Zero Exponent

Learn why any nonzero number to the power of zero equals 1, and how to handle expressions like $5x^0$ and $(-3)^0$.

Intermediate20 min

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