Pattern: 5³=125, 5²=25, 5¹=5, 5⁰=1. Each time ÷5.

What is 5⁰? Why Any Number to Power 0 is 1

Question

What is $5^0$ ? Prove that any non-zero number raised to the power 0 equals 1.

Solution — Step by Step

List out descending powers of 5 and see what happens each time:

5^3 = 125, \quad 5^2 = 25, \quad 5^1 = 5

Notice anything? Each step down, we divide by 5.

We go from $5^1 = 5$ to $5^0$ by dividing once more:

5^0 = 5^1 \div 5 = 5 \div 5 = 1

The pattern forces the answer — there’s no choice involved.

The law says: $\dfrac{a^m}{a^n} = a^{m-n}$ , as long as $a \neq 0$ .

Set $m = n$ :

\frac{5^3}{5^3} = 5^{3-3} = 5^0

But $\dfrac{5^3}{5^3} = \dfrac{125}{125} = 1$ . So $5^0 = 1$ .

Replace 5 with any non-zero number $a$ :

\frac{a^n}{a^n} = a^{n-n} = a^0 \quad \text{and} \quad \frac{a^n}{a^n} = 1

Therefore $a^0 = 1$ for all $a \neq 0$ .

Answer: $5^0 = 1$

Why This Works

The exponent laws aren’t arbitrary rules — they’re shorthand for repeated multiplication. When we write $5^3 \div 5^3$ , we’re cancelling three 5s in the numerator against three 5s in the denominator. The result is $1$ , always.

The pattern approach (dividing by the base each time you reduce the exponent) is actually the most intuitive way to see this. It shows that $a^0 = 1$ isn’t a “definition we made up” — it’s the only value that keeps the pattern consistent.

This result holds for every non-zero base: $7^0 = 1$ , $(-3)^0 = 1$ , $\left(\dfrac{2}{5}\right)^0 = 1$ . The base doesn’t matter. Only $0^0$ is left undefined — that’s a separate story for higher classes.

Alternative Method

Use the product rule: multiply $a^0$ by $a^1$ and see what base forces the answer.

We know $a^m \times a^n = a^{m+n}$ . So:

a^0 \times a^1 = a^{0+1} = a^1

This means $a^0 \times a = a$ .

Divide both sides by $a$ (valid since $a \neq 0$ ):

a^0 = \frac{a}{a} = 1

Same answer, different route. This method is useful in Class 9–10 when you need to justify the zero-exponent rule formally, not just observe the pattern.

Common Mistake

Many students write $5^0 = 0$ . This is wrong — and easy to see why. The exponent tells you how many times to multiply. Zero doesn’t mean “the answer is zero”; it means you’ve applied the base zero times, which by the division law gives 1. Confusing the exponent with the result is the trap here.

A second mistake: thinking $0^0 = 1$ by the same rule. Don’t extend it there — $0^0$ is indeterminate (undefined), and your NCERT book won’t ask it, but examiners sometimes use it as a distractor in MCQs in Classes 9 and 10.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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