Powers and their properties

A power is shorthand for repeated multiplication: instead of 2·2·2·2·2 we write 2⁵. The laws of indices underlie roots, exponential and logarithmic equations, standard form and the simplification of algebraic expressions. In physics very large and very small quantities (3·10⁸ m/s, 1.6·10⁻¹⁹ C) are written with powers too. This lesson also explains why the definitions are chosen as they are and when the laws must not be used.

§1. Natural powers

a is the base, n the index (exponent), aⁿ the power. The second power is called the “square” and the third the “cube”: a square of side a has area a², a cube has volume a³.
The sign rules follow from multiplication:

• Any power of a positive number is positive.
• An even power of a negative number is positive, an odd power negative: (−3)² = 9, (−3)³ = −27. Reason: negative factors cancel in pairs.
• −3² = −9 (the power applies only to 3), (−3)² = 9. Without brackets the power comes first, then the minus.
• 0ⁿ = 0, 1ⁿ = 1 (for every natural n).

§2. The main laws

Each law follows directly from the definition. For example, a³·a² = (a·a·a)·(a·a) = a⁵ — count the factors: 3 + 2 = 5. (a³)² = a³·a³ = a⁶ — here the indices are multiplied.
The laws work only with the same base (multiplying and dividing) or the same index. There is no law for sums: 2³ + 2⁴ cannot become 2⁷. Instead, take out a common factor: 2³ + 2⁴ = 2³(1 + 2) = 24.
Rewriting with a common base helps a lot: 4 = 2², 8 = 2³, 27 = 3³, 0.25 = 2⁻².

Worked example

Calculate (2²⁰ + 2¹⁸)/(5·2¹⁷).

1. Take 2¹⁸ out of the numerator: 2¹⁸(2² + 1) = 5·2¹⁸.
2. The fraction becomes 5·2¹⁸/(5·2¹⁷).
3. The 5 cancels, 2¹⁸ : 2¹⁷ = 2.
4. Answer: 2.

§3. Zero, negative and fractional indices

These definitions are not arbitrary — they are forced by keeping the laws true. aⁿ : aⁿ = 1, while the law gives aⁿ⁻ⁿ = a⁰, so a⁰ = 1. Likewise a⁰ : aⁿ = a⁻ⁿ, i.e. a⁻ⁿ = 1/aⁿ. For a fractional index we need (a^(1/n))ⁿ = a¹ = a, hence a^(1/n) = ⁿ√a.
0⁰ and negative powers of zero are undefined (they would mean dividing by zero).
Examples: (0.5)⁻³ = 2³ = 8, 27^(2/3) = (∛27)² = 9, 16^(−3/4) = 1/(⁴√16)³ = 1/8.

Worked example (from the official sample)

Find the value of (4n²/m)² · k/(m²n²) : k³/(mn)³ · mk²/n³ for n = −3, m = −2, k = 3.

1. Simplify first (operations left to right): (4n²/m)² = 16n⁴/m².
2. · k/(m²n²) → 16n²k/m⁴.
3. : k³/(m³n³) → 16n²k·m³n³/(m⁴k³) = 16n⁵/(mk²).
4. · mk²/n³ → 16n².
5. n = −3: 16·9 = 144.

Simplifying first makes the arithmetic easy and safe: m and k were not even needed.

§4. Comparing powers

To compare, make the bases or the indices equal.

For example, 0.5³ < 0.5² because 0.125 < 0.25: a number below one shrinks when multiplied by itself.

Worked example

Compare 5⁴⁰ and 4⁵⁰.

1. The HCF of the indices is 10: 5⁴⁰ = (5⁴)¹⁰ = 625¹⁰.
2. 4⁵⁰ = (4⁵)¹⁰ = 1024¹⁰.
3. 625 < 1024, so 5⁴⁰ < 4⁵⁰.

§5. The last digit

The last digit of a power depends only on the last digit of the base and repeats in cycles:

Method: divide the index by the cycle length; with remainder r take the r-th term of the cycle, with remainder 0 take the last one.

Worked example

Find the last digit of 3¹⁰⁰.

1. Cycle: 3, 9, 7, 1 (length 4).
2. 100 = 4·25, remainder 0 — the fourth term.
3. Answer: 1.

§6. Standard form

n is the order of magnitude. For large numbers n is positive, for numbers below one it is negative: 0.000 52 = 5.2·10⁻⁴ (the decimal point moved 4 places right).
When multiplying, multiply the leading numbers and add the powers of ten, then restore standard form: (3·10⁵)·(4·10⁻²) = 12·10³ = 1.2·10⁴.
When dividing, divide the leading numbers and subtract the powers: (6·10⁸) : (2·10⁻³) = 3·10¹¹. To compare two positive numbers in standard form, compare the orders first: the larger order wins; with equal orders compare the leading numbers. For example, 9.9·10⁴ < 1.1·10⁵, even though 9.9 > 1.1.

§7. Simplifying expressions with powers

In a complicated expression this order almost always works:

1. Rewrite every base as a power of a prime (9 = 3², 27 = 3³, 0.2 = 5⁻¹).
2. Write roots and reciprocals as powers (√a = a^(1/2), 1/aⁿ = a⁻ⁿ).
3. For each base, add and subtract the indices separately.
4. Evaluate at the end.

The method works even when the index contains a letter: the letters cancel and the answer is a number. Exactly this skill is needed later for exponential equations such as 2ˣ = 32: once both sides have the same base, the indices can be equated (2ˣ = 2⁵ → x = 5).

Worked example

Simplify 9ⁿ·27 : 3²ⁿ⁺¹.

1. Use base 3: 9ⁿ = 3²ⁿ, 27 = 3³.
2. Numerator: 3²ⁿ·3³ = 3²ⁿ⁺³.
3. Divide: 3²ⁿ⁺³ : 3²ⁿ⁺¹ = 3²ⁿ⁺³⁻²ⁿ⁻¹ = 3².
4. Answer: 9 — independent of n.

Worked example

Calculate (8^(2/3) + 81^(1/4))·2⁻¹.

1. 8^(2/3) = (∛8)² = 2² = 4.
2. 81^(1/4) = ⁴√81 = 3.
3. 2⁻¹ = 1/2.
4. (4 + 3)·1/2 = 3.5.

Check yourself

1. Calculate 2⁵·2³ : 2⁶.

2. Calculate (3⁻²)⁻¹·3⁰.

3. Calculate 27^(2/3).

4. Which is larger: 2³⁰ or 3²⁰?

5. The last digit of 7²⁰²⁶?

6. Calculate (0.5)⁻³.

7. Calculate (3⁷ − 3⁵)/3⁵.

8. The last digit of 2²⁰²⁵?

Common mistakes

• Adding indices when adding powers: 2³ + 2⁴ ≠ 2⁷.
• Adding indices in (aᵐ)ⁿ — they must be multiplied: (2³)² = 2⁶, not 2⁵.
• Reading a negative index as a negative number: 2⁻³ = 1/8, not −8.
• Giving 0⁰ a value — it is undefined.
• Confusing −3² with (−3)².
• Adding indices of powers with different bases: writing 2³·3² as 6⁵.