Powers in algebra

Take a sheet of paper and fold it in half. Now it has 2 layers. Fold it again and it has 4. After 5 folds it has 32 layers, because every fold doubles what you had: 2 · 2 · 2 · 2 · 2 = 32. Writing the same number again and again is slow, so we use a short way to write it: 2⁵2⁵. This is called a power. A few simple rules let you work with powers in algebra quickly, without writing everything out.

What a power means

In 2⁵2⁵, the 2 is the base: the number that is multiplied. The small 5 is the exponent (or the power): it tells you how many times the base appears. So 2⁵2⁵ means five 2s multiplied together, and x³x³ means x · x · x.

Every rule below comes from this one idea. If you forget a rule, write the powers out as long multiplications and count the letters. You will get the rule back.

Multiplying powers: add the exponents

What is x³ x⁴x³ x⁴? Write both powers out. x³x³ is three xs and x⁴x⁴ is four xs. Multiplied together, that is 3 + 4 = 7 of them in a row.

Dividing powers: subtract the exponents

A fraction line means divide. Look at (x⁸)/(x²)x⁸x². On top there are eight xs, and on the bottom there are two. Each x on the bottom cancels one x on the top, because x/x = 1xx = 1. Two of the eight are gone, so six are left.

A power of a power: multiply the exponents

Now look at (x³)²(x³)². The small 2 outside the bracket means the whole bracket is multiplied by itself: (x³)² = x³ x³(x³)² = x³ x³. By the first rule, that is x^(3 + 3) = x⁶x^{3 + 3} = x⁶. Two groups of three xs make 3 · 2 = 6.

A power of a bracket with a number in it

In (2x)³(2x)³, everything inside the bracket is cubed, the number 2 as well as the x. Write it out: (2x)³ = 2x · 2x · 2x(2x)³ = 2x · 2x · 2x. Put the numbers together and the letters together: 2 · 2 · 2 = 8 and x · x · x = x³x · x · x = x³.

When the unknown is the exponent

Sometimes x sits up in the exponent, as in 2ˣ = 322^x = 32. The question is: how many 2s must you multiply to get 32? The plain way is to count. Start at 2 and keep multiplying by 2 until you reach the number.

1. Write the powers of the base in a row: the base, then the base times itself, and so on.
2. Count how many steps it takes to reach the number.
3. That count is x.

If the exponent is a small expression, like x + 1, find the whole exponent first, then solve a simple equation for x.

Undoing a square

In x² = 144x² = 144 we want a number that, multiplied by itself, gives 144. That is the square root of 144, written √144√144. Since 12 · 12 = 144, the answer is 12. Strictly, −12 works too, because (−12)(−12) = 144 as well. That is why the question tells you that x is positive: then the answer is only 12.

1. Get x²x² alone on one side, the same way you solve any equation.
2. Take the square root of the number on the other side.
3. Check by multiplying your answer by itself.