When we have negative powers (e.g. 7^{-2}) we reciprocate it and make the power positive. Reciprocating a number means that we divide 1 by the number. Here are a few examples of numbers/ unknowns and their reciprocals:

> 4 has the reciprocal ^{1}/_{4}

> 10 has the reciprocal ^{1}/_{10}

> 25 has the reciprocal ^{1}/_{25}

*> x* has the reciprocal ^{1}/_{x}

> 21*y* has the reciprocal ^{1}/_{21y}

When we reciprocate a fraction, we flip the fraction; the original numerator becomes the new denominator, and the original denominator becomes the new numerator. Here are a few examples:

> ^{3}/_{4} has the reciprocal ^{4}/_{3}

> ^{2}/_{7} has the reciprocal ^{7}/_{2}

After we have found the reciprocal for a negative power, we then make the power positive and complete the calculation.

**Example 1**

What is:

We have a negative power; the power is -2. Therefore, we need to reciprocate it and change the power from a negative to a positive. This gives us:

7 now has a positive power of 2, which means that we square 7.

Therefore, we can write the answer as ^{1}/_{49}.

The key rule with a negative power is to reciprocate it and then change the power to a positive power.

**Example 2**

What is:

The power is negative (-3), which means that we need to reciprocate this and then make the power positive. This gives us:

The next step is to work out what 4^{3} is, which is 4 multiplied by itself 3 times; 4 x 4 x 4.

The answer is ^{1}/_{64}.

**Fractional Negative Power**

We are now going to have a look at an example whereby we have a fraction to a negative power. We answer questions like this in the same way as all of the previous negative power questions; we reciprocate the base (number or fraction) and change the power from a negative power to a positive power. When we reciprocate a fraction, the original numerator becomes the new denominator, and the original denominator becomes the new numerator.

**Example 3**

What is:

We have a negative power, so the first step is to flip the fraction and change the power from a negative to a positive power.

There are two different ways that we can finish off the rest of this question.

**Way 1**

The first way is to multiply the base (^{2}/_{3}) by itself 4 times. When we multiply fractions, we multiply straight across. The working is shown below:

The answer is ^{16}/_{81}

**Way 2**

The second way is to bring the power outside the bracket (4) in for each of the individual terms. When we raise a power to another power, we multiply the powers together. Currently, the 2 and the 3 have a power of 1. The working is shown below:

The next step is to take each of the numbers out of index form.

The answer is ^{16}/_{81}.

Both of these methods give the same answer, so choose the method that you prefer.