Indices

This section covers Indices and their uses in Algebra including: How to divide and multiply algebraic expressions using indices. Find roots using indices and understand the core laws of Indices: Product of Powers Law, Quotient of Powers Law, Power of a Power Law, Zero Exponent Law and The Negative Exponent Law. 

This video shows a guide to indices and powers. Multiplying and dividing indices, raising indices to a power and using standard form are explained.

The Laws of Indices

Product of Powers Law

When multiplying powers with the same base, add the indices:

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

Quotient of Powers Law

When dividing powers with the same base, subtract the indices:

$$\frac{a^m}{a^n} = a^{m-n}$$

Power of a Power Law

When raising a power to another power, multiply the indices:

$$\left(a^m\right)^n = a^{m \times n}$$

Zero Exponent Law

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

$$a^0 = 1 \quad \text{(for } a \neq 0\text{)}$$

Negative Exponent Law

A negative exponent means the reciprocal of the base raised to the positive exponent:

$$a^{-m} = \frac{1}{a^m} \quad \text{(for } a \neq 0\text{)}$$

Examples

$\sqrt{\frac{x^6}{y^{12}}} = \left(\frac{x^6}{y^{12}}\right)^\frac{1}{3} = \frac{x^2}{y^4}$

$(a^{4}b^{10}) \div (ab^2) = a^{4-1} \times b^{10-2} = a^{3}b^{8}$

$(xy^3)^4 = x^{4}y^{12}$

If $x^2 = a^2 - 16$ then $x = \sqrt{a^2 - 16}$ not $a-4$: Always be careful when square rooting expressions. When in doubt, try with simple numbers, for example: $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ (not $3 + 4 = 7$)

More information on Index Forms, Roots and Laws can be found here.

Advanced indices

You can find more advanced Indices content here