Once you"ve learned about negative numbers, you can also learn about negative powers. A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x–2" (pronounced as "ecks to the minus two") just means "x2, but underneath, as in 1/(x2)".

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Write x–4 using only positive exponents.

I know that the negative exponent means that the base, the x, belongs on the other side of the fraction line. But there isn"t a fraction line!


To fix this, I"ll first convert the expression into a fraction in the way that any expression can be converted into a fraction: by putting it over "1". Of course, once I move the base to the other side of the fraction line, there will be nothing left on top. But since anything can also be regarded as being multiplied by 1, I"ll leave a 1 on top.

Here"s what it looks like:


Once I no longer needed the "1" underneath (to create the fraction), I omitted it, because I had the variable expression underneath, and the "times one" doesn"t change anything.

Write x2 / x–3 using only positive exponents.

Only one of the terms has a negative exponent. This means that I"ll only be moving one of these terms. The term with the negative power is underneath; this means that I"ll be moving it up top, to the other side of the fraction line. There already is a term on top; I"ll be using exponent rules to combine these two terms.

Once I move that denominator up top, I won"t having anything left underneath (other than the "understood" 1), so I"ll drop the denominator.


The negative power will become just "1" once I move the base to the other side of the fraction line. Anything to the power 1 is just itself, so I"ll be able to drop this power once I"ve moved the base.


Make sure you understand why the "2" above does not move with the variable: the negative exponent is only on the "x", so only the x moves..

I"ve got a number inside the power this time, as well as a variable, so I"ll need to remember to simplify the numerical squaring.

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Unlike the previous exercise, the parentheses meant that the negative power did indeed apply to the three as well as the variable.

Write (-5x-1)/(y3) using only positive powers.

The "minus one" power on the x means that I"ll need to move that x to the other side of the fraction line. But the "minus" on the 5 means only that the 5 is negative. This "minus" is not a power, so it doesn"t say anything about moving the 5 anywhere!