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We have seen square roots before, however, we will go into a bit more depth here.

Playlist Steps for Simplifying Radical Expressions

Also, we will generalize this concept and introduce nth roots. The definition of the principal (non-negative) square root follows.

Simplify.

Not all square roots work out so nicely. If trying to simplify a square root with a radicand that is not a perfect square, we can find the exact answer using the following properties.

Here A and B are non-negative real numbers and B is not equal to zero. Approximations are made using a calculator.

Give the exact answer and approximate to the nearest hundredth.

This idea can be extended to any positive integer index n.

For 3rd roots, or cube roots, the question is “what raised to the 3rd power will produce the given number?" For example,

because

Simplify.

Remember that if the radicand of an odd root is negative then the result will be a negative real number. If the radicand of an even root is negative then the number is not real.

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Up to this point we have been solving quadratic inequalities. The technique involving sign charts extends to solving polynomial inequalities of higher degree.

Solving Quadratic, Rational, and Polynomial Inequalities Playlist

Solve.

Step 1: Determine the critical numbers, which are the roots or zeros in the case of a polynomial inequality.

Step 2: Create a sign chart.

Step 3: Use the sign chart to answer the question.

The last problem shows that not all sign charts will alternate. Do not take any shortcuts and test each interval.

Rational inequalities are solved using the same technique. The only difference is in the critical numbers. It turns out that the y-values may change from positive to negative at a restriction. So we will include the zeros of the denominator in our list of critical numbers.

Solve.

Tip: Always use open dots for critical numbers that are also zeros of the denominator, or restrictions. This reminds us that they are restrictions and should not be included in the solution set even if the inequality is inclusive.