Indices and Laws of Indices

Introduction to Indices

Indices (singular: index), also known as powers or exponents, provide a shorthand way of writing repeated multiplication of the same number. Instead of writing 2 × 2 × 2 × 2, we can write 2⁴, which is read as "2 to the power of 4" or "2 raised to the power 4." In this expression, 2 is called the base and 4 is called the index (exponent or power). Indices make it much easier to write, read, and calculate with very large or very small numbers, and they follow specific rules (called the laws of indices) that simplify calculations involving powers.

Meaning of an Index

When a number, called the base, is multiplied by itself a certain number of times, the result can be expressed using an index. In general, aⁿ means a multiplied by itself n times, where a is the base and n is the index. For example, 5³ = 5 × 5 × 5 = 125. The index tells us how many times the base appears as a factor in the multiplication.

Basic Terminology

In the expression 3⁵, the number 3 is the base, and the number 5 is the index, exponent, or power. The whole expression 3⁵ is read as "3 to the power of 5" or simply "3 to the 5th," and it equals 3 × 3 × 3 × 3 × 3 = 243. Special cases have their own names: a number raised to the power 2 is called "squared" (e.g., 7² is "7 squared"), and a number raised to the power 3 is called "cubed" (e.g., 4³ is "4 cubed").

First Law of Indices: Multiplication

When multiplying two powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128. This rule works because multiplying powers of the same base simply combines the total number of times the base is multiplied by itself.

Second Law of Indices: Division

When dividing two powers with the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625. This rule works because dividing cancels out the common factors of the base, leaving only the difference in how many times the base remains multiplied.

Third Law of Indices: Power of a Power

When raising a power to another power, multiply the indices: (aᵐ)ⁿ = aᵐˣⁿ. For example, (2³)² = 2³ˣ² = 2⁶ = 64. This can be understood as squaring the value of 2³ (which is 8), so (2³)² = 8² = 64, matching the result from the rule.

Zero Index

Any non-zero number raised to the power of zero equals 1: a⁰ = 1 (where a ≠ 0). For example, 7⁰ = 1, and 100⁰ = 1. This rule follows logically from the division law: since aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any number divided by itself equals 1, it follows that a⁰ must equal 1.

Negative Indices

A negative index indicates the reciprocal of the base raised to the corresponding positive index: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8. Negative indices are useful for expressing very small numbers and for working with fractions and reciprocals in a compact form.

Fractional Indices (Roots)

A fractional index represents a root of the base. Specifically, a^(1/n) means the nth root of a. For example, a^(1/2) means the square root of a (√a), and a^(1/3) means the cube root of a (∛a). For example, 9^(1/2) = √9 = 3, and 8^(1/3) = ∛8 = 2. More generally, a^(m/n) means the nth root of a raised to the power m, or equivalently, (a^(1/n))ᵐ.

Power of a Product and Power of a Quotient

When raising a product to a power, each factor is raised to that power separately: (ab)ⁿ = aⁿbⁿ. For example, (2 × 3)² = 2² × 3² = 4 × 9 = 36, which matches 6² = 36. Similarly, when raising a quotient (fraction) to a power, both the numerator and denominator are raised to that power: (a/b)ⁿ = aⁿ/bⁿ. For example, (2/3)² = 2²/3² = 4/9.

Applying the Laws of Indices Together

Many problems require applying more than one law of indices in combination. For example, to simplify (2³ × 2⁴) ÷ 2⁵, first apply the multiplication law to the numerator: 2³ × 2⁴ = 2⁷, then apply the division law: 2⁷ ÷ 2⁵ = 2². The result simplifies to 2² = 4. Being comfortable combining these laws step by step is essential for solving more complex algebraic expressions involving indices.

Common Mistakes with Indices

Students often make errors when working with indices, such as incorrectly adding or multiplying the bases instead of just combining the indices (remember, the base stays the same in multiplication and division laws), forgetting that a negative index means a reciprocal rather than a negative number, and applying the laws of indices to expressions with different bases, where these laws do not directly apply unless the bases are first converted to a common base.

Real-Life Applications of Indices

Indices are widely used in real life and other areas of mathematics and science: expressing very large numbers, such as distances in space or populations, in standard form; calculating compound interest and population growth, which involve repeated multiplication over time; representing digital storage sizes in computing (such as kilobytes, megabytes, and gigabytes, based on powers of 2); and expressing scientific measurements, such as the sizes of atoms or the speed of light, using powers of ten.

Summary

Indices provide a compact way of expressing repeated multiplication of the same base number. The key laws of indices include: adding indices when multiplying powers of the same base, subtracting indices when dividing powers of the same base, multiplying indices when raising a power to another power, and understanding that any non-zero number raised to the power zero equals 1. Negative indices represent reciprocals, and fractional indices represent roots. Mastering these laws allows students to simplify complex expressions efficiently and forms an essential foundation for algebra and advanced mathematics.

Indices Beyond the Classroom

Beyond pure calculation, indices form the mathematical backbone of many real-world phenomena that grow or shrink very rapidly, a pattern known as exponential growth or decay. Population growth, the spread of information or diseases, radioactive decay, and the accumulation of compound interest on savings all follow patterns that can only be properly described using indices. Recognising this connection helps students appreciate why the laws of indices, though sometimes appearing abstract, are among the most practically important tools in the whole of mathematics.

Practising the Laws of Indices

Confidence with indices comes from repeated practice applying each law separately before combining them in more complex expressions. Students should practise simplifying expressions involving the same base with multiplication, division, and powers of powers, and should also practise converting between negative indices, fractional indices, and their equivalent root or reciprocal forms. Working through a wide variety of examples, and always checking answers by substituting small numbers back into the original expression, helps build the fluency needed to apply the laws of indices quickly and accurately in examinations and in later algebra topics such as simplifying algebraic fractions and solving exponential equations, both of which are frequently encountered in senior secondary mathematics.

Worked Examples

Example 1: Simplify 3⁴ × 3². Since the bases are the same, add the indices: 3⁴⁺² = 3⁶ = 729.

Example 2: Simplify 7⁸ ÷ 7⁵. Since the bases are the same, subtract the indices: 7⁸⁻⁵ = 7³ = 343.

Example 3: Simplify (5²)³. Multiply the indices: 5²ˣ³ = 5⁶ = 15,625.

Example 4: Evaluate 4⁻². A negative index gives a reciprocal: 4⁻² = 1/4² = 1/16.

Example 5: Evaluate 27^(1/3). A fractional index of 1/3 means the cube root: ∛27 = 3.

Student Exercise

Solve the following problems, showing all your working:

1. Simplify 2⁵ × 2³.
2. Simplify 6⁹ ÷ 6⁴.
3. Simplify (3²)⁴.
4. Evaluate 5⁰.
5. Evaluate 2⁻⁴.
6. Evaluate 16^(1/2).
7. Evaluate 64^(1/3).
8. Simplify (2 × 5)³.
9. Simplify (3/4)².
10. Simplify (4³ × 4²) ÷ 4⁴.