Index laws tell you how to combine powers of the same base. Negative and fractional powers are the versions students most often get stuck on. A negative power means “take the reciprocal”, and a fractional power means “take a root”. Once those two ideas are secure, the rest is the same set of laws you already know.
This lesson opens indices, roots and standard form. It also supports algebra later, because the same laws work on letters.
What are the index laws?
Write a for any non-zero base, and m and n for the powers.
| Law | Example |
|---|---|
| am × an = am+n | 34 × 32 = 36 |
| am ÷ an = am−n | 56 ÷ 52 = 54 |
| (am)n = amn | (23)2 = 26 |
| a0 = 1 | 90 = 1 |
| a−n = 1/an | 4−2 = 1/16 |
| a1/n = the n-th root of a | 81/3 = 2 |
The negative-power law comes from the division law. 22 ÷ 25 = 2−3, and written out in full it is (2 × 2) / (2 × 2 × 2 × 2 × 2) = 1/8. So 2−3 = 1/8.
How do I handle a fraction raised to a negative power?
Flip the fraction, then drop the minus sign. (2/3)−2 = (3/2)2 = 9/4. This is faster than working out 1/(2/3)2 and gives the same answer.
For a fractional power m/n, take the n-th root and then the m-th power. Taking the root first is kinder on the arithmetic, because the numbers stay small.
Worked example
Find the exact value of (4/9)−3/2.
Step 1, deal with the minus sign. Flip the fraction and remove the minus: (4/9)−3/2 = (9/4)3/2.
Step 2, deal with the denominator of the power. The 2 means a square root: √(9/4) = 3/2.
Step 3, deal with the numerator. The 3 means cube it: (3/2)3 = 27/8.
Step 4, check a different way. √(4/9) = 2/3, and (2/3)3 = 8/27. The minus sign means reciprocal, so the value is 27/8. Both routes agree.
Answer: 27/8. Leave it as an exact fraction unless the question asks for a decimal.
The mistake to watch for
The usual slip is to treat the minus sign as if it makes the number negative.
Mistaken working: 2−3 = −8
The student multiplied 2 × 2 × 2 and then attached the minus sign from the power.
The correction: the minus sign in the power belongs to the direction of the calculation, not to the answer. 2−3 = 1/23 = 1/8.
A quick test helps: a positive base raised to any power, positive or negative, is always positive. If your answer for 2−3 is below zero, the working has gone wrong.
Check yourself
Try these without a calculator, then open each answer.
1. Find the value of 4−2.
Show answer
4−2 = 1/42 = 1/16.
2. Find the value of (2/5)−2.
Show answer
Flip the fraction: (5/2)2 = 25/4. The answer is 25/4, which is 6.25.
3. Find n if 2n = 1/32.
Show answer
32 = 25, so 1/32 = 2−5. Therefore n = −5.
Where this leads next
Next, see how a negative power turns up in measurements in converting a small measurement to standard form. When you want mixed questions, use the indices, roots and standard form practice set. The non-calculator working trainer lets you check exact steps without leaning on a calculator.
Some students can recite the laws but choose the wrong one under pressure. A teacher can spot that quickly in online one-to-one Mathematics tuition.