Laws of Indices

Indices (also called exponents or powers) are a shorthand way of writing repeated multiplication. Instead of writing 2 x 2 x 2 x 2, we write 2 to the power 4. In Grade 11, you will use these laws constantly in algebra, logarithms, and even science.

The Seven Key Laws

Let a and b be non-zero numbers, and m and n be any real numbers.

1. Product Rule: When multiplying with the same base, add the exponents. a^m x a^n = a^(m+n)

2. Quotient Rule: When dividing with the same base, subtract the exponents. a^m / a^n = a^(m-n)

3. Power Rule: When raising a power to another power, multiply the exponents. (a^m)^n = a^(m x n)

4. Zero Exponent: Any non-zero number raised to the power zero equals 1. a^0 = 1

5. Negative Exponent: A negative exponent means take the reciprocal. a^(-n) = 1/a^n

6. Product of Bases: (ab)^n = a^n x b^n

7. Fractional Exponent: a^(1/n) = the nth root of a

Worked Examples

Example 1: Simplify 2^3 x 2^5

Using the Product Rule: 2^3 x 2^5 = 2^(3+5) = 2^8 = 256

Example 2: Simplify (3^2)^4 / 3^5

First, apply the Power Rule: (3^2)^4 = 3^8. Then, Quotient Rule: 3^8 / 3^5 = 3^3 = 27

Example 3: Evaluate 5^(-2)

Using the Negative Exponent rule: 5^(-2) = 1/5^2 = 1/25

Example 4: Simplify 8^(1/3)

This means the cube root of 8. Since 2 x 2 x 2 = 8, the answer is 2.

Common Mistakes to Avoid

• 2^3 x 3^2 is NOT 6^5 -- the bases must be the same to use the product rule
• (2 + 3)^2 is NOT 2^2 + 3^2 -- exponents do not distribute over addition
• 2^0 = 1, not 0

Did You Know? In Nepal, population growth is often modeled using exponential expressions. If a city grows by 2% each year, after n years the population is P x (1.02)^n -- that is the power of indices in real life!

Key Takeaways

• Same base multiplication: add exponents; division: subtract exponents
• Power of a power: multiply exponents
• Zero exponent always gives 1; negative exponent means reciprocal
• Fractional exponents represent roots