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Radicals Study Guide (page 2)

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Updated on Oct 3, 2011

Solving Equations with Radicals

Now that we know how to work with radicals, we can use them to help us solve equations. So far, we have used addition, subtraction, multiplication, and division to solve equations. Sometimes, we will need to raise both sides of an equation to a power, or take a root of both sides of an equation in order to find our answer.

Example

x2 = 64

How can we find the value of x? Addition, subtraction, multiplication, and division cannot help us. However, we can get x alone on the left side of the equation if we take the square root of both sides of the equation. Why the square root? Because the exponent of x is 2. If the exponent of x was 5, we would take the fifth root of x to get x alone. Remember the tip we learned earlier: If the exponent and root of a base are the same, the term can be simplified to just the base or its absolute value.

x2 = √64

The square root of x2 is the absolute value of x and the square root of 64 is 8, since (8)(8) = 64. We are left with x = 8. This isn't our only answer, though. It's true that the square root of 64 is 8, but there is another number that, when squared, equals 64: –8. Remember, (–8)(–8) = 64.

Our answers are x = 8, –8.

Example

4x2 = 144

To solve for x, we start by dividing both sides of the equation by 4.

x2 = 36

Now, we can take the plus and minus square roots of 36. Because (6)(6) = 36 and (–6)(–6) = –36, our answers are x = 6, –6.

We can also use exponents to help us simplify radicals. To remove the radical symbol from an equation, raise it to an exponent that is equal to the root of the radical.

Example

4r = 3

Because the fourth root of r is equal to 3, raise both sides of the equation to the fourth power to remove the radical symbol.

(4r)4 =34

r = 81

Find practice problems and solutions for these concepts at Radicals Practice Questions.

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