Indices and Surds
Senior 2 · Mathematics
Indices and Surds
Lesson 1 of 5
22 min
Suppose Mugisha shares a piece of school news with 3 friends. Each of them tells 3 more people, and each of those tells 3 more. After four rounds of sharing, 3 × 3 × 3 × 3 = 81 new people have heard the news. Writing long products like this is slow, so mathematics uses a short form: 34.
Powers are everywhere in science and technology. Area is measured in m2 and volume in m3. Phone memory is counted in powers of 2, and scientists write the distance from the Earth to the Sun with powers of 10. The laws in this lesson let you simplify such expressions quickly, and every later topic in algebra depends on them.
For any number a and any positive whole number n,
an = a × a × … × a (n factors).
The number a is the base. The small raised number n is the index (plural: indices). It is also called the exponent or the power. We read 34 as "3 to the power 4", and 34 = 3 × 3 × 3 × 3 = 81.
Brackets matter. (−2)4 = (−2)(−2)(−2)( − 2) = 16, but −24 means −(24) = −16, because the index applies only to the 2.
Multiplying powers of the same base. 23 × 24 = (2 × 2 × 2) × (2 × 2 × 2 × 2). That is seven 2s multiplied together, so 23 × 24 = 27. The indices were added: 3 + 4 = 7.
Dividing powers of the same base. 56 ÷ 52: two of the six 5s on top cancel with the two 5s underneath, leaving 54. The indices were subtracted: 6 − 2 = 4.
A power of a power. (32)3 = 32 × 32 × 32 = 36. The indices were multiplied: 2 × 3 = 6.
The zero index. 73 ÷ 73 = 1, because any non-zero number divided by itself is 1. The subtraction law gives 73 − 3 = 70. Both answers must agree, so 70 = 1.
Negative indices. 23 ÷ 25 = 8 ÷ 32 = 14 = 122. The subtraction law gives 23 − 5 = 2−2. So 2−2 = 122. A negative index means "one over" the positive power.
| Law | In symbols (a, b ≠ 0) | Example |
|---|---|---|
| Product | am × an = am + n | x5 × x2 = x7 |
| Quotient | am ÷ an = am − n | y9 ÷ y4 = y5 |
| Power of a power | (am)n = amn | (p3)4 = p12 |
| Power of a product | (ab)n = anbn | (2k)3 = 8k3 |
| Power of a quotient | (a ÷ b)n = an ÷ bn | (x ÷ 3)2 = x2 ÷ 9 |
| Zero index | a0 = 1 | 20260 = 1 |
| Negative index | a−n = 1 ÷ an | 10−3 = 1 ÷ 1000 |
The product and quotient laws work only when the bases are the same. 23 × 32 = 8 × 9 = 72, and it cannot be written as a single power.
Simplify (2x3y2)3 × 4x−2 ÷ (8x5y).
Step 1: remove the bracket. Each factor inside is cubed: 23 = 8, (x3)3 = x9, (y2)3 = y6. So the bracket is 8x9y6.
Step 2: multiply by 4x−2. Numbers: 8 × 4 = 32. Powers of x: 9 + ( − 2) = 7. The result is 32x7y6.
Step 3: divide by 8x5y. Numbers: 32 ÷ 8 = 4. Powers of x: 7 − 5 = 2. Powers of y: 6 − 1 = 5.
Answer: 4x2y5.
Evaluate 2−3 × 42 ÷ 80 without a calculator.
Write every number as a power of 2: 42 = (22)2 = 24, and 80 = 1. Then 2−3 × 24 ÷ 1 = 2−3 + 4 = 21 = 2.
Tip
Exam technique. Before you apply any law, rewrite every number with the same base (4 = 22, 8 = 23, 9 = 32, 27 = 33). Then remember the pattern: multiply → add the indices, divide → subtract, bracket → multiply. You never multiply the bases: 23 × 24 is 27, not 47.
Work each item on paper before you read the solution.
Simplify 34 × 35 ÷ 36 and evaluate it.
Solution: 34 + 5 − 6 = 33 = 27.
Simplify (x4)3 ÷ x5.
Solution: (x4)3 = x12, and x12 ÷ x5 = x7.
Evaluate 50 + 4−1 + 2−2.
Solution: 50 = 1, 4−1 = 14 and 2−2 = 14. The total is 1 + 14 + 14 = 32.
Simplify (3a2b)2 × 2ab3.
Solution: (3a2b)2 = 9a4b2. Multiply by 2ab3: 9 × 2 = 18, a4 + 1 = a5, b2 + 3 = b5. Answer: 18a5b5.
Write 16 × 8 ÷ 32 as a single power of 2, then evaluate it.
Solution: 24 × 23 ÷ 25 = 24 + 3 − 5 = 22 = 4.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
