OCR GCSE Maths: Laws of Indices

GCSE Maths: Laws of Indices

The laws of indices are a set of rules that govern how we work with powers. They allow us to simplify expressions involving exponents and make calculations easier. Here's a breakdown of the key rules:

Multiplication Rule

When multiplying powers with the same base, add the exponents.

$x^m \times x^n = x^{m+n}$

Example:

$x^3 \times x^5 = x^{3+5} = x^8$

Division Rule

When dividing powers with the same base, subtract the exponents.

$x^m \div x^n = x^{m-n}$

Example:

$x^7 \div x^2 = x^{7-2} = x^5$

Power-of-a-Power Rule

When raising a power to another power, multiply the exponents.

$(x^m)^n = x^{m \times n}$

Example:

$(x^4)^3 = x^{4 \times 3} = x^{12}$

Zero Power Rule

Any non-zero number raised to the power of zero equals 1.

$x^0 = 1$ (where $x ? 0$)

Example:

$5^0 = 1$

Negative Exponent Rule

A number raised to a negative exponent is equal to its reciprocal raised to the positive version of that exponent.

$x^{-n} = \frac{1}{x^n}$

Example:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Fractional Exponent Rule

A fractional exponent indicates a root. The numerator represents the power, and the denominator represents the root.

$x^{m/n} = \sqrt[n]{x^m}$

Example:

$x^{2/3} = \sqrt[3]{x^2}$

Practice Problems

Simplify the following expressions using the laws of indices:

1. $x^2 \times x^4$
2. $y^6 \div y^3$
3. $(z^5)^2$
4. $2^{-2}$
5. $8^{2/3}$

Answers:

1. $x^6$
2. $y^3$
3. $z^{10}$
4. $\frac{1}{4}$
5. 4

Remember: These laws are essential for working with powers in algebra and other areas of mathematics. Practice applying them to different expressions to build your understanding and confidence.