# Negative powers

$10^{-4}$is equal to...

## Negative powers

If you're beginning to understand exponential expressions, it's time to take a look at negative powers. What happens if you take a number and raise it to the power of a negative number? What is 4 multiplied by itself minus three times? To understand this, it's good to remember how to divide exponential expressions. To do that, you subtract the powers.

The numerator's power minus the denominator's power. And if the denominator’s power is larger than the numerator’s power, you get a negative power. But let's back up a little. An exponential expression is a repeated multiplication. Write out this division as a repeated multiplication.

And simplify. Look at the first and the last line. Four to the power of minus three equals 1 divided by four to the power of three. This is not a coincidence, this is a general rule, which you can learn by heart: An exponential expression with a negative power is the same as 1 divided by the same expression with a positive power. When you see a minus in front of the power, you can simply remove it and take 1 divided by that exponential expression.

Four to the power of minus three is 1 divided by 4 raised to the power of 3. And four to the power of three is 64. So four to the power of minus three is one sixty-fourth Or approximately 0.02 When working with orders of magnitude, negative powers are even easier. Take 10 to the power of minus two. That is 1 / 10 squared, which is one hundredth.

And that is 0.01 Test it with 10 to the power of -4. That is one ten-thousandth. A shortcut here is to let the power show you the number of places to move the decimal point to the left. 10 to the power of minus 6 is equal to ‘one’ with the decimal point moved 1, 2, 3, 4, 5, 6 places to the left. One millionth.

In other words, there are two things to remember: An exponential expression with a negative power is the same as ‘one’ divided by the same expression - but with a positive power. And then the special case when the base is ten: Start with one and move the decimal point as many places to the left as the power indicates.