Change of Base Formula

The change of base formula is very important because most calculators do not have a log to any base button.

YouTube Playlist: Logarithms and their Graphs

This formula allows us to calculate any logarithm given only two bases on our calculator.

Apply the change of base formula as follows.

We could choose any base. However, it is wise to choose base e or 10 because there is a button for them on most calculators.

Evaluate using a calculator and round off to the nearest thousandth.

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Radicals Including Square and Cube Roots

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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Complex Rational Expressions

It turns out that we have all the tools necessary to simplify complex algebraic fractions.

Rational Expressions and Equations Playlist on YouTube

The numerator and denominator of these rational expressions contain fractions and look very intimidating.  We will outline two methods for simplifying them.

Method 1: Obtain a common denominator for the numerator and denominator, multiply by the reciprocal of the denominator, then factor and cancel if possible.

Method 2: Multiply the numerator and denominator of the complex fraction by the LCD of all the simple fractions then factor and cancel if possible.

To illustrate what happened after we multiplied by the LCD we could add an extra step.

For the following solved problems, both methods are used. Choose whichever method feels most comfortable for you.

Simplify using method 1. Simplify using method 2.

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