Equations with exponents and root expressions

​When you're solving an equation and you need to remove terms and factors from either side, you can use the fact that mathematical operations have opposites. To get rid of plus five for example, you add minus five to both sides. And to get rid of one-third, you simply multiply both sides by three. The mathematical operations have opposites, and you use this when working with equations. An example of another pair of opposites you can use is powers and roots.

Let's say you have to construct a circle with the area 113 centimeters squared and you want to know what is it's radius. You know that an area of a circle is Pi multiplied by the radius squared. We know the area. Plug it in the equation. Divide both sides by Pi to have the radius, which is the unknown variable we want on its own, on one side.

We can simplify the right side to remove Pi, and we can simplify the left side to 113 divided by Pi. Pi is not an unknown variable. It is a constant who's value we already know. 113 divided by Pi or 3.14 is approximately 36. But what do we do now? ​ ​The radius squared is 36.

So what is the radius? Well the opposite of a power, is a root. So we take the square root of the entire left side and the square root of the entire right side. The equation is still an equality. Now comes the mathematical magic.

Look at the right side. The square root of R to the power of 2 equals R. Square root tells you what number multiplied by itself equals the number we have here and to the power of 2, tells you, "Take this number multiplied by itself." So they cancel each other out. All that remains is the square root of 36 and that is 6. Since 6 times 6 is 36.

R = 6. ​A circle with an area of 113 centimeters squared has a radius of 6 centimeters. ​ ​This works the other way around as well. If you're solving an equation and get stuck when it looks like this, the square root of X equals 8. Then you take the equation to the power of two to eliminate the square root sign. Take both sides squares - that is, multiply them by themselves - but wait, you know that you have to do the same thing with the left and right sides. And here, we're multiplying the left side by the square root of X and the right side by 8.

We are not doing the same thing, are we? Actually, we are. Look at the line above. It says the square root of X equals 8. So when we square both sides we multiply them by the same value then we just need to simplify.

The square root of X to the power of two is X. And 8 squared is 64. X equals 64. ​ ​Remember these two tips when solving equations. When you encounter a squared variable, eliminate the square by taking the square root and when you encounter a variable under the square root sign, square it.