Here we look at equations that have roots in the problem. As you might expect, to clear a root we can raise both sides to an exponent. So to clear a square root we can rise both sides to the second power. To clear a cubed root we can raise both sides to a third power. There is one catch to solving a problem with roots in it, sometimes we end up with solutions that do not actually work in the equation. This will only happen if the index on the root is even, and it will not happen all the time. So for these problems it will be required that we check our answer in the original problem. If a value does not work it is called an extraneous solution and not included in the final solution.
When solving a radical problem with an even index: check answers!
Even index! We will have to check answers. Square both sides, simplify exponents Solve Subtract from both sides Divide both sides by Need to check answer in original problem Multiply Add Square root True! It works! Our Solution
Even index! We will have to check answers Rise both sides to fourth power Solve Subtract from both sides Divide both sides by Need to check answer in original problem Multiply Add Take root False, extraneous solution No Solution Our Solution
In this example one can also notice at the very beginning that in the left hand side is always nonnegative, no mater what the value of is, hence it will never be equal to the righ hand side .
Even index! We will have to check solutions Isolate radical by subtracting from both sides Squarebothsides Evaluate exponents, recall Reorder terms Make equation equal zero Subtract and from both sides Factor Set each factor equal to zero or Solve each equation or Need to check answers in original problem Check first Add Take root Add False, extraneous root Check Add Take root Add True! It works Our Solution
The above example illustrates that as we solve we could end up with an term or a quadratic. In this case we remember to set the equation to zero and solve by factoring. We will have to check both solutions if the index in the problem was even. Sometimes both values work, sometimes only one, and sometimes neither works
If there is more than one square root in a problem we will clear the roots one at a time. This means we must first isolate one of them before we square both sides.
Even index! We will have to check answers Isolate first root by adding to both sides Square both sides Evaluate exponents Solve Subtract from both sides Divide both sides by Need to check answer in original Multiply Subtract Take roots Subtract True! It works Our Solution
When there is more than one square root in the problem, after isolating one root and squaring both sides we may still have a root remaining in the problem. In this case we will again isolate the term with the second root and square both sides. When isolating, we will isolate the term with the square root. This means the square root can be multiplied by a number after isolating.
Even index! We will have to check answers Isolate first root by adding to both sides Square both sides Evaluate exponents, recall Isolate the term with the root Subtract and from both sides Square both sides Evaluate exponents Make equation equal zero Subtract from both sides Factor Set each factor equal to zero or Solve Add to both sides of second equation or Need to check answers in original Check first Take roots Subtract True! It works Check Add Take roots SubtractTrue! It works or Our Solution
Even index! We will have to check answers Isolate the first root by adding Square both sides Evaluate exponents Combine like terms Isolate the term with radical Subtract and from both sides Square both sides Evaluate exponents Distribute Make equation equal zero Subtract and from both sides Factor Set each factor equal to zero or Solve or Check solutions in original Check first Take roots Subtract False, extraneous solution Check Add Take roots Subtract True! It works Our Solution