Algebraic Index Laws for the TMUA

Index Notation

Algebraic terms often involve powers, written in the form $a^n$. Here, $a$ is known as the base and $n$ is the index (the plural is indices) or the power. This notation represents repeated multiplication: for instance, $a \times a \times a \times a \times a$ is written more compactly as $a^5$.

The Law of Multiplication

To multiply powers of the same base, add the indices together:

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

When terms include numerical coefficients, such as the 5 in $5p$, these coefficients must be multiplied as normal numbers. For example, to simplify $5p^2q \times 2pq^2$, multiply the coefficients 5 and 2 to get 10, then apply the addition law to the powers of $p$ and $q$ separately, resulting in $10p^3q^3$.

Worked Example: Multiplication

Question: Simplify $4a^2 \times 2a^3$.

Solution: Multiply the numerical coefficients and add the indices for the common base $a$:

$4 \times 2 \times a^{2+3} = 8a^5$

The Law of Division

To divide powers of the same base, subtract the divisor's index from the dividend's index:

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

Numerical coefficients are divided first. For example, $20p^3q^2 \div 2p^2q$ is simplified by dividing 20 by 2 to get 10, then subtracting the powers to get $p^{3-2}q^{2-1}$, which results in $10pq$.

Worked Example: Division

Question: Simplify $12a^6 \div 2a^3$.

Solution: Divide the coefficients and subtract the indices:

$(12 \div 2) \times a^{6-3} = 6a^3$

Raising Terms to a Further Power

To raise a term to a further power, multiply the indices:

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

It is vital that the powers of all numbers and letters within the term are multiplied by the external index. For example, $(4ab^2)^3 = 4^3 a^3 b^{2 \times 3} = 64a^3b^6$.

Worked Example: Raising a Power

Question: Simplify $(2p^3)^5$.

Solution: Raise the coefficient to the power and multiply the indices of the base:

$2^5 \times p^{3 \times 5} = 32p^{15}$

Zero and Unitary Indices

There are two essential identities involving the indices 0 and 1:

1. Any non-zero base raised to the power of zero equals 1: $a^0 = 1$.
2. Any base raised to the power of one is simply itself: $a^1 = a$.

Additionally, 1 raised to any power remains 1.

Powers of Fractions

When a fraction is raised to a power, the index applies to both the numerator and the denominator:

$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

Worked Example: Fractional Bases

Question: Simplify $\left(\frac{2p^2}{3q^3}\right)^4$.

Solution: Apply the power 4 to every element in the numerator and denominator:

$\frac{(2p^2)^4}{(3q^3)^4} = \frac{2^4 p^{2 \times 4}}{3^4 q^{3 \times 4}} = \frac{16p^8}{81q^{12}}$

Negative Powers

A base raised to a negative power can be written as 1 over the base to the positive power:

$a^{-m} = \frac{1}{a^m}$

Worked Example: Negative Indices

Question: Simplify $(4q^3)^{-4}$.

Solution: Treat the entire term as a base raised to a negative power, which results in its reciprocal raised to the positive power:

$\frac{1}{(4q^3)^4} = \frac{1}{256q^{12}}$

Fractional Powers

Fractional indices represent roots. The denominator of the fraction identifies the order of the root:

1. The power $\frac{1}{2}$ is the square root: $a^{\frac{1}{2}} = \sqrt{a}$.
2. The power $\frac{1}{3}$ is the cube root: $a^{\frac{1}{3}} = \sqrt[3]{a}$.

In the general case where the numerator is not 1:

$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Worked Example: Simplifying Fractional Powers

Question: Simplify $(25y^4)^{\frac{1}{2}}$.

Solution: Apply the square root to the coefficient and the variable:

$25^{\frac{1}{2}} (y^4)^{\frac{1}{2}} = 5 y^{4 \times \frac{1}{2}} = 5y^2$

Worked Example: Negative and Fractional Indices

Question: Simplify $(8a^{-3})^{\frac{1}{3}}$.

Solution: Apply the cube root to the coefficient and the base:

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