Indices and Surds

Definition

An index (plural: indices) is the power, or exponent, of a number. For example, $a^3$ has an index of 3.

A surd is an irrational number that can be expressed with roots, such as $\sqrt{2}$ or $\sqrt[5]{19}$.

Technique

The manipulation of indices and surds can be a powerful tool for evaluating and simplifying expressions.

Let's start with some basic rules for operations with indices:

$\begin{aligned} x^m \times x^n &= x^{m+n} \\ \frac{x^m}{x^n} &= x^{m-n} \\ x^{-n} &= \frac{1}{x^n} \\ (x^m)^n &= x^{mn} \\ (xy)^{n} &= x^n y^n \\ \left(\frac{x}{y}\right)^m &= \frac{x^m}{y^m} \end{aligned}$

Surds are just numbers with fractional indices, e.g. $\sqrt[3]{7^2} = 7^{2/3}$. Any operation with indices can be applied to surds, and indices and surds are related through this rule:

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

This allows us to group numbers together into forms that can be more convenient. Here are a couple examples:

Simplify: $2^5 \times 4^3$

$\begin{aligned} 2^5 \times 4^3 &= 2^5 \times (2^2)^3 \\ &= 2^5 \times 2^6 \\ &= 2^{11} \end{aligned} _\square$

Simplify: $\displaystyle \frac{(a^2)^4 b^7}{ab^{-2}}$

$\begin{aligned} \frac{(a^2)^4 b^7}{ab^{-2}} &= \frac{a^8 b^7}{ab^{-2}} \\ &= \frac{a^7 b^7}{b^{-2}} \\ &= a^7 b^9 \end{aligned} _\square$

Sometimes surds will appear in the dominator of an expression. You can rationalize the denominator by applying the following technique to a fraction of the form $\frac{a}{b+\sqrt{c}}$:

$\frac{a}{b+\sqrt{c}} \left( \frac{b-\sqrt{c}}{b-\sqrt{c}} \right) = \frac{ab-a\sqrt{c}}{b^2 -c}$ For example:

Simplify: $\displaystyle \frac{1}{2+\sqrt{5}}$

$\begin{aligned} \frac{1}{2+\sqrt{5}} &= \frac{1}{2+\sqrt{5}} \left( \frac{2-\sqrt{5}}{2-\sqrt{5}} \right) \\ &= \frac{2-\sqrt{5}}{2^2 -5} \\ &= -2 + \sqrt{5} \end{aligned} _\square$

Application and Extensions

If you write the prime factorization of$\sqrt[5]{32^8}$, what is the sum of indices of the factors?

If you recognize that $32 = 2^5$, the answer falls quickly into place:

$\sqrt[5]{32^8} = (2^5)^\frac{8}{5} = 2^8$

So, the sum of the indices is simply 8. $_\square$

If $3^{x-y} = 81$ and $3^{x+y} = 729$, what is $x$?

Multiply the two expressions together to get the $y$'s to cancel out: $\begin{aligned} 3^{x-y} \times 3^{x+y} &= 81 \times 729 \\ 3^{x-y+x+y} &= 3^4 \times 3^6 \\ 3^{2x} &= 3^{10} \\ x &= 5 \end{aligned} _\square$