Meaning of indices

In an expression such as $a^r$, the number or expression $a$ is the base and $r$ is the index. For example, in $x^{3/2}$, the base is $x$ and the index is $\frac{3}{2}$.

For positive integer indices, the meaning is repeated multiplication:

The laws of indices are designed so that this repeated-multiplication pattern keeps working when the indices are rational numbers such as $\frac{1}{2}$, $\frac{3}{4}$, or $-\frac{2}{3}$.

For this lesson, when a variable has a fractional exponent, assume the base is positive unless a restriction is stated. For example, write $x>0$ when using $x^{1/2}$, $x^{3/2}$, or $x^{-2/3}$ as general real-valued algebra. When dividing by powers of a base, the base must also be non-zero.

Worked example 1

Identify the base and index in each expression.

1. $5^{2/3}$
2. $(2x-1)^{-4}$
3. $p^{1/2}q^3$

Solution

In $5^{2/3}$, the base is $5$ and the index is $\frac{2}{3}$.

In $(2x-1)^{-4}$, the whole bracket $2x-1$ is the base and the index is $-4$. The brackets matter: $(2x-1)^{-4}$ is not the same as $2x-1^{-4}$.

In $p^{1/2}q^3$, there are two bases. The base $p$ has index $\frac{1}{2}$, and the base $q$ has index $3$. Index laws must be applied separately to each base.

The three core laws

The Pearson specification names three core laws. They are laws for powers with the same base.

The first law adds indices because the repeated factors are being put together. For instance,

The second law subtracts indices because division cancels common factors:

The third law multiplies indices because a power is being repeated:

These examples use integer indices so the structure is visible. The specification requires the same laws to be used for all rational exponents, so the same moves apply to expressions such as $x^{1/2}\times x^{3/2}$ and $(x^{2/3})^6$.

The quotient law also explains the zero index. If $a\ne 0$, then

but the quotient law gives

So $a^0=1$ for any non-zero base $a$.

Worked example 2

Simplify each expression, giving your answer using positive indices where possible. Assume $x>0$.

Solution

For the product, the bases are both $x$, so add the indices:

For the quotient, the bases are both $x$, so subtract the indices:

For the power of a power, multiply the indices:

Using positive indices,

So the three simplified answers are

The three core laws Continued

The most common error is to use the right operation with the wrong law. Multiplication of powers means add indices, not multiply them. A power raised to a power means multiply indices, not add them.

Fractional and negative indices

Fractional indices connect powers to roots. The denominator of the fraction tells you the root; the numerator tells you the power.

This is the equivalence required by the specification. It lets you choose the easier order. For $16^{3/4}$, it is easier to take the fourth root first:

For $27^{2/3}$, take the cube root first:

A negative index means reciprocal:

So a negative fractional index combines both ideas:

Worked example 3

Evaluate exactly:

Solution

For $81^{3/4}$, take the fourth root first:

For $32^{-2/5}$, deal with the negative index by making a reciprocal:

Then use the fifth root:

So

For $\left(\frac{1}{16}\right)^{-3/4}$, the negative index first changes the base to its reciprocal:

Now use the fourth root:

Explain It Back

Use this as a self-explanation check after the section above. It is for diagnosing what you can already explain, not for learning new material from scratch.

That example is a useful stress test: the answer is positive because the negative index creates a reciprocal, not a negative number.

Simplifying with common bases

Index laws only combine powers with the same base. Sometimes the base is already the same, as in $x^{1/2}x^{5/2}$. Sometimes you need to rewrite numbers using a common base first.

For example, $8$, $16$ and $32$ can all be written as powers of $2$:

This makes an expression such as $8^{2/3}\times 16^{1/2}$ much easier to simplify:

Now apply the power law:

Then multiply:

Worked example 4

Simplify fully:

Solution

Rewrite every number as a power of $5$:

Substitute these into the expression:

Apply the power law:

So the expression becomes

Combine the powers in the numerator:

Then divide by $5=5^1$:

Therefore

Simplifying with common bases Continued

Do not force a common base when there is no useful common base. For example, $2^3\times 5^2$ is already exact; it is $8\times25=200$, not $10^5$.

Algebraic expressions

In algebraic simplification, apply index laws separately to each base. If an expression contains both $x$ and $y$, collect the powers of $x$ together and the powers of $y$ together.

Worked example 5

Simplify

where $x>0$ and $y>0$. Give your answer using positive indices.

Solution

First simplify the numerical coefficient:

Now collect the powers of $x$. Since the powers are being divided, subtract the indices:

Now collect the powers of $y$:

So the expression is

Using positive indices,

Algebraic expressions Continued

When checking your answer, look at each base separately. A fully simplified answer should not leave a negative index unless the question allows it.