Mathematics • Interactive Lesson
Indices (Exponents)
Understand the base and exponent, discover each law visually, and practise using correctly formatted mathematical expressions.
24
base 2 • exponent 4
Read an exponent correctly
What does 24 mean?
The exponent 4 does not mean “2 × 4”. It means that the base 2 appears as a factor four times.
2×
2×
2×
2→
16
Read: “two to the power of four” or “two to the fourth power”.
Positive exponent
34 = 3 × 3 × 3 × 3 = 81
The base is repeatedly multiplied.
Zero exponent
a0 = 1 when a ≠ 0
A non-zero base raised to zero equals 1.
Negative exponent
a−n = 1an
A negative exponent creates a reciprocal.
Fractional exponent
amn = ⁿ√(am)
The denominator gives the root; the numerator gives the power.
The ten laws of indices
1Product rule
am × an = am+n
23 × 24 = 27 = 128
3 factors+4 factors→7 factors
2Quotient rule
aman = am−n
5753 = 54 = 625
7 factors−3 cancelled→4 remain
3Power of a power
(am)n = amn
(32)4 = 38 = 6561
2 per group×4 groups→8 factors
4Power of a product
(ab)n = anbn
(2 × 3)4 = 24 × 34
(ab)n→anbn
5Power of a quotient
abn
=
anbn
b ≠ 0
253
=
2353
=
8125
power to top+power to bottom
6Zero index
a0 = 1 a ≠ 0
70 = 1 and (−5)0 = 1
amam= 1 and exponent m−m = 0
7Negative index
a−n = 1an
2−3 = 123 = 18
negative power→ reciprocal →positive power
8Fractional index
amn = ⁿ√(am)
823 = ∛(82) = 4
denominator n = rootnumerator m = power
9Index of 1
1n = 1
110 = 1 and 1−5 = 1
1 × 1 × 1 …→1
10Positive index of 0
0n = 0 when n > 0
02 = 0 and 05 = 0
0 × 0 × 0 …→0
Explore the power of a quotient
Build your own example
The exponent must be applied separately to the numerator and denominator.
253
→
2353
→
8125
Power calculator
24 = 16
Interactive assessment
Question 1 of 10
Score: 0
Quick review
Quick values
| Expression | Value |
|---|---|
| 25 | 32 |
| (−2)3 | −8 |
| 50 | 1 |
| 12−2 | 4 |
| 912 | 3 |
Common mistakes
- Add indices only when multiplying powers with the same base.
- Subtract indices only when dividing powers with the same base.
- Keep brackets around a negative base.
- Do not use the zero-index rule when the base is zero.
A reliable method
1Identify the operation
2Check the base
3Apply one law
4Simplify and check
Full worked steps
Product rule
23 × 24
- Confirm the bases are equal.
- Use the product rule.
- Add the exponents: 3 + 4 = 7.
- Write 27.
- Evaluate: 128.
Quotient rule
5753
- Confirm the bases are equal.
- Use the quotient rule.
- Subtract: 7 − 3 = 4.
- Write 54.
- Evaluate: 625.
Negative exponent
2−3
- Negative exponent means reciprocal.
- Rewrite as 123.
- Evaluate 23 = 8.
- Final answer: 18.
Power of a quotient
253
- Apply the exponent to the numerator and denominator.
- Calculate 23 = 8.
- Calculate 53 = 125.
- Final answer: 8125.
Fractional exponent
823
- The denominator 3 means cube root.
- ∛8 = 2.
- The numerator 2 means square.
- 22 = 4.