Roots and Radicals (41/36) -- Intermediate Algebra

Roots and Radicals Simplify Expressions with Roots Learning Objectives By the end of this section, you will be able to: - Simplify expressions with roots - Estimate and approximate roots - Simplify variable expressions with roots Before you get started, take this readiness quiz. Simplify Expressions with Roots In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example, Square Square Root Notice (−13)2 = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, which denotes the positive square root of m. The positive square root is also called the principal square root. We also use the radical sign for the square root of zero. Because Notice that zero has only one square root. We know that every positive number has two square roots and the radical sign indicates the positive one. We write If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, Simplify: ⓐ ⓑ ⓐ ⓑ Simplify: ⓐ ⓑ ⓐⓑ 15 Simplify: ⓐ ⓑ ⓐ 10 ⓑ Can we simplify Is there a number whose square is Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to The square root of a negative number is not a real number. Simplify: ⓐ ⓑ ⓐ ⓑ Simplify: ⓐ ⓑ ⓐ not a real number ⓑ Simplify: ⓐ ⓑ ⓐⓑ not a real number So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots. Let’s review some vocabulary first. The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube. It will be helpful to have a table of the powers of the integers from −5 to 5. See (Figure). Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2 to help you see this. We will now extend the square root definition to higher roots. Just like we use the word ‘cubed’ for b3, we use the term ‘cube root’ for We can refer to (Figure) to help find higher roots. Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers. When n is an even number and - then is a real number. - then is not a real number. When n is an odd number, is a real number for all values of a. We will apply these properties in the next two examples. Simplify: ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ Simplify: ⓐ ⓑ ⓒ ⓐ 3 ⓑ 4 ⓒ 3 Simplify: ⓐ