Number · Number Operations · Lesson 9

The Zero Index: Why Any Number to the Power 0 Is 1

What could $5^{0}$ mean, when you cannot write 5 as a factor zero times? In this lesson you will discover, using patterns and division of powers, that any natural number raised to the power 0 equals 1. You will then use this fact, together with the rule for dividing powers, to simplify expressions.

NaCCA B7.1.2.3.3 ⏱ about 40 min

By the end you will be able to

✔Continue a pattern of powers to show that $2^{0}$ = 1.
✔Divide powers of the same base using $a^{m} \div a^{n} = a^{m - n}$ .
✔Explain why any natural number with index 0 has value 1, e.g. 27 ÷ 27 = $3^{3} \div 3^{3} = 3^{0}$ = 1.
✔Use the zero index to simplify expressions and write numbers in expanded form.

Key words

Zero index: An index (exponent) of 0, as in $7^{0}$ . For any natural number $a$ , $a^{0}$ = 1.
Natural number: A counting number: 1, 2, 3, 4, …
Quotient: The answer to a division. The quotient of 27 ÷ 27 is 1.
Division law of indices: When dividing powers of the same base, subtract the indices: $a^{m} \div a^{n} = a^{m - n}$ .

Curriculum reference

Content standard: B7.1.2.3 Demonstrate understanding and the use of powers of natural numbers in solving problems.

Indicator: Show that the value of any natural number with zero as its exponent or index is 1 and use it to solve problems.

Starter

Sharing 27 mangoes among 27 pupils

Mr Mensah brings 27 ripe mangoes from his farm at Nkawkaw and shares them equally among 27 pupils. Everyone knows each pupil gets 27 ÷ 27 = 1 mango. But 27 = $3^{3}$ , so the sharing is $3^{3} \div 3^{3}$ . Kojo works it out with powers and gets $3^{0}$ . So is $3^{0}$ equal to 0, to 3, or to 1? Today you will find out, and the answer surprises many people.

1. Follow the pattern down to index 0

Look at the powers of 2 written from the largest index to the smallest. Each time the index goes down by 1, we remove one factor of 2, which means we divide by 2. If we keep the same pattern going one more step, we reach index 0.

Each step down divides by the base
Power of 2	Value	Power of 3	Value	Power of 10	Value
$2^{4}$	16	$3^{4}$	81	$1 0^{4}$	10,000
$2^{3}$	8	$3^{3}$	27	$1 0^{3}$	1,000
$2^{2}$	4	$3^{2}$	9	$1 0^{2}$	100
$2^{1}$	2	$3^{1}$	3	$1 0^{1}$	10
$2^{0}$	1	$3^{0}$	1	$1 0^{0}$	1

▶ Notice that each row is half the one above. Halving 2 counters leaves 1 counter, not zero counters.

Whatever the base, the last step divides the base by itself: 2 ÷ 2 = 1, 3 ÷ 3 = 1, 10 ÷ 10 = 1. So in every column the power with index 0 has value 1.

2. Dividing powers of the same base

To divide powers with the same base, write each one as repeated factors and cancel the common factors. For example:

$2^{5} \div 2^{2} = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2} = 2 \times 2 \times 2 = 2^{3}$

Two of the 2s on top cancel with the two 2s at the bottom, leaving 5 − 2 = 3 factors. This works every time, so we get the division law of indices: $a^{m} \div a^{n} = a^{m - n}$ . (In the same way, when you multiply powers of the same base you add the indices: $a^{m} \times a^{n} = a^{m + n}$ .)

Using the division law
Division	Subtract indices	Answer	Check with values
$2^{5} \div 2^{2}$	5 − 2 = 3	$2^{3}$	32 ÷ 4 = 8 ✓
$3^{4} \div 3^{1}$	4 − 1 = 3	$3^{3}$	81 ÷ 3 = 27 ✓
$1 0^{6} \div 1 0^{2}$	6 − 2 = 4	$1 0^{4}$	1,000,000 ÷ 100 = 10,000 ✓

3. Why any number to the power 0 equals 1

Now divide a power by itself. We know that any number divided by itself is 1, so 4 ÷ 4 = 1. But 4 = $2^{2}$ , so using the division law:

$\frac{4}{4} = 2^{2} \div 2^{2} = 2^{2 - 2} = 2^{0}$

The same division gave both 1 and $2^{0}$ , so ** $2^{0}$ = 1**. Try it with 27: $\frac{27}{27}$ = 1, and also $3^{3} \div 3^{3} = 3^{3 - 3} = 3^{0}$ . Therefore $3^{0}$ = 1. This happens for every natural number $a$ : $a^{m} \div a^{m}$ is 1 and is also $a^{0}$ .

4. Using the zero index to solve problems

Once you know $a^{0}$ = 1, replace every power with index 0 by 1, then follow the usual order of operations (powers first, then × and ÷, then + and −).

$7^{0} + 9^{0}$ = 1 + 1 = 2
6 × $4^{0}$ = 6 × 1 = 6
$2^{3} \times 3^{0}$ = 8 × 1 = 8
$1 0^{2} - 1 0^{0}$ = 100 − 1 = 99

The zero index also completes expanded form using powers of 10. The ones place has value 1 = $1 0^{0}$ , so 3,405 = 3 × $1 0^{3}$ + 4 × $1 0^{2}$ + 0 × $1 0^{1}$ + 5 × $1 0^{0}$ .

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Verifying with 27

Use

\frac{27}{27}

to show that

3^{0}

= 1.

Show the solution

Step 1Any number divided by itself is 1, so $\frac{27}{27}$ = 1.
Step 2Write 27 as a power: 27 = 3 × 3 × 3 = $3^{3}$ .
Step 3Use the division law: $3^{3} \div 3^{3} = 3^{3 - 3} = 3^{0}$ .
Step 4Both answers come from the same division, so $3^{0}$ = 1.

Answer: $3^{0}$ = 1

Example 2

Dividing powers

Simplify

2^{5} \div 2^{5}

and

2^{5} \div 2^{3}

, giving each answer as a power and as a number.

Show the solution

Step 1 $2^{5} \div 2^{5} = 2^{5 - 5} = 2^{0}$ = 1. (Check: 32 ÷ 32 = 1 ✓)
Step 2 $2^{5} \div 2^{3} = 2^{5 - 3} = 2^{2}$ = 4. (Check: 32 ÷ 8 = 4 ✓)

Answer: $2^{0}$ = 1 and $2^{2}$ = 4

Example 3

Simplifying with zero indices

Simplify

5^{0} + 4^{0} \times 6

Show the solution

Step 1Replace the zero powers: $5^{0}$ = 1 and $4^{0}$ = 1.
Step 2The expression becomes 1 + 1 × 6.
Step 3Multiply first: 1 × 6 = 6.
Step 4Then add: 1 + 6 = 7.

Answer: 7

Example 4

Expanded form with powers of 10

Write 3,405 in expanded form using powers of 10.

Show the solution

Step 13,405 = 3 thousands + 4 hundreds + 0 tens + 5 ones.
Step 2Thousands = $1 0^{3}$ , hundreds = $1 0^{2}$ , tens = $1 0^{1}$ , ones = 1 = $1 0^{0}$ .
Step 33,405 = 3 × $1 0^{3}$ + 4 × $1 0^{2}$ + 0 × $1 0^{1}$ + 5 × $1 0^{0}$ .
Step 4Check: 3,000 + 400 + 0 + 5 = 3,405 ✓

Answer: 3 × $1 0^{3}$ + 4 × $1 0^{2}$ + 0 × $1 0^{1}$ + 5 × $1 0^{0}$

Summary

Going down the powers of a number, each step divides by the base; one step below index 1 gives index 0 and value 1.
Division law: $a^{m} \div a^{n} = a^{m - n}$ (same base only). Multiplication law: $a^{m} \times a^{n} = a^{m + n}$ .
A number divided by itself is 1, and $a^{m} \div a^{m} = a^{0}$ , so $a^{0}$ = 1.
$a^{0}$ is never 0 and never $a$ : $7^{0}$ = 1.
In expanded form, the ones place is $1 0^{0}$ = 1.

Sharing 27 mangoes among 27 pupils

1. Follow the pattern down to index 0

2. Dividing powers of the same base

3. Why any number to the power 0 equals 1

4. Using the zero index to solve problems

Worked examples

Verifying with 27

Dividing powers

Simplifying with zero indices

Expanded form with powers of 10

Summary

Practice: 15 questions