Exponents and Square Roots

To square a number means to multiply that number times itself.  For example:

The number 5 is called the base and the integer 2 is called the exponent.

Exponential Notation 

Another way to read the above example is "5 raised to the second power."

Simplify.

It is important to point out the last two solved problems, what is different?

 

The square root can be thought of as the opposite of squaring a number.  In other words, if we want the square root of 16 the question is "what number squared gives 16?"  Actually there are two answers to this question, 4 and −4 because:

Hence, there is a technical distinction here. When we are asked for the square root of numbers we mean the principal (non-negative) square root.

Simplify.

If A and B are positive and real the following property is useful for simplifying square roots:

Look for the largest perfect square factor and the apply this property as follows:

Video Examples on YouTube:

Complex Numbers

Up to this point, we have been working exclusively with real numbers.  Now we shed this limitation and allow for a much broader range of problems.

Complex Numbers Playlist on YouTube

With these definitions, we have greatly expanded our space of numbers.  Notice that any real number is also a complex number, for example 5 = 5 + 0i.  Here the real part is 5 and the imaginary part is 0.  Next we consider powers of i.

Simplify.

Adding and subtracting complex numbers is just a matter of adding like terms. Be sure to use the order of operations and add real and imaginary parts separately.

Add or subtract.

We have to use a bit of caution when multiplying complex numbers.  First, we will run into i^2 often. In this case, we will replace them all with -1.  In addition, the property

is true only when either A or B is non-negative; therefore, simplify using the imaginary unit before multiplying.

Tip: Make use of the imaginary unit if the radicand is negative before trying to simplify.

Multiply.

Multiplying complex numbers often requires the distributive property.

Dividing complex numbers requires techniques similar to rationalizing the denominator.

Divide.

When a complex number is in the denominator, multiply numerator and denominator by its conjugate.

Video Examples on YouTube: