12 1 Rational Exponents 12 2 Simplifying Radicals

12. 1 Rational Exponents 12. 2 Simplifying Radicals

• A square root is the number in which we multiply by itself 2 times to get the number under the root symbol. • The number under the root symbol is called the Radicand. • this is a symbol that represents the number that we multiply by itself 2 times to get 64. What is that number?

• Likewise this is the symbol for the number that we multiply by itself two times to get 15. • When there is not a small number out in front of the root symbol then it is an understood 2. Which we call the square root. • The number that is out front as part of the symbol is called the INDEX.

• The bottom of page 829 has a list of the most commonly used perfect squares, cubes, and 4 th roots.

• We cannot take the square root (or any even root) of a negative number. That is because you can never take two numbers (that are actually the same number) that are both negative and multiply them together and get a negative number.

• However, you can take the cube root (or any odd root) of a negative number because three negative numbers multiplied together will result in a negative number.

• Sometimes there will be variables that are placed underneath the radical symbol. The general practice is to simplify the radicals, therefore we will try to reduce the exponents under a radical by drawing a connection between the INDEX of the radical and the exponent of the RADICANDS.

Simplify

Radicals can be written using exponents • If x is a real number and n is a positive integer greater than 1, then

Examples to Simplify

• Sometimes it is easier to write expressions using rational exponents and use properties of exponents (usually power to power) to simplify.

• The rational exponent theorem allows us to pull apart rational exponents (reverse of power to a power rule). For example.

Simplify the following

12. 2 Simplifying Radicals When simplifying radicals it is best to deal with the numerical parts separate. We will conduct what is called the PRIME FACTORIZATION of the radicand. -That means we will break the radicand down to the factors that multiply together to give the radicand. We will then group them based on the INDEX.

Now we can do the same thing with variables under the radical. You will want to re-write the variables that have higher exponents in multiples that relate directly to the INDEX For example…

Now put it all together inside one problem

Simplifying Fractions under a radical Break it up into two separate radicals. Numerator and denominator Then simplify each radical as far as possible.

• When you do this and a radical is left in the denominator you need to do what is called “Rationalizing the denominator”. To do this we multiply by a convenient value of 1. (that value will always be whatever the denominator is) • After you multiply you may have to reduce the radicals again.

• So far we have assumed that all variables were nonnegative (0 or positive). Now we want to think about them “possibly” being negative. • Remember though we do not know what the variable represents so it could be positive or it could be negative. • This shows that the number being squared could be 3 or it could be -3 and still get the same 9

• So lets look at we do not know whether x represents a positive or negative number. So when we bring it out of the radical we are not sure whether it should be positive or negative. • If we are assuming that we do not know anything about the numbers being nonnegative then whenever a value comes out of the radical we have to put it inside absolute value bars.