Unit 1: Operations

Radicals
Radicals are a fun concept, but in order to understand them, you must know a few fundamental properties.
So, here's a few of properties you should memorize:

• ⁿ√(ab) = ⁿ√(a) x ⁿ√(b)
• ⁿ√(a/b) = ⁿ√(a) / ⁿ√(b)
• ⁿ√(a) = a^1/2
• ^m√(ⁿ√(a)) = ^mn√(a)

You can use these properties to simplify radicals, or solve radical equations.

As with all mathematical elements, radicals have parts. The radical itself can be of multiple powers (i.e. square roots, cube roots), so the root that you are trying to find is called the index. The value under the radical symbol is called the radicand.

In order to add or subtract radicals, their radicands must be the same. Add or subtract the coefficients, and keep the common radicand in order to find the sum or difference. Remember, you can never add or subtract radicals with different radicands.

In order to multiply or divide radicals, use the above properties. Multiplying the coefficients yeilds the resulting coefficient. As for the radical part, they remain seperate unless they share an index. You can always combine radicals with the same index, but you cannot place a square root and a cube root under the same radical.

Solving equations with radicals involves using any of the above properties to find a variable.  Remember, you must always follow these rules, or Mr.Sihler will throw you into math jail. When you find a result, always check for extraneous, or invalid, solutions. Always, always, always plug your answers back into the original equation to make sure that they are valid. Often, radicals create extra solutions that do not exist.

Operations with Complex Numbers and i
Complex numbers can often be confusing and difficult to understand, but the basic idea originates from the concept of the square roots of negative numbers. Complex numbers involve both real numbers, and imaginary ones. Imaginary numbers share a common factor, i, which represents the square root of -1. i and its multiples must always be treated as seperate from the real portion of the complex number, similarly to radicals.

Adding or subtracting complex numbers is a relatively simple concept. Real numbers can be added together, and like radicals, imaginary numbers can also be grouped together. Simply combine the coefficients, and leave i as it is. Remember that complex numbers are still numbers on their own, and must be treated as such.

Multiplying complex numbers is slightly more complicated. You can use the distributive property to combine real and imaginary numbers, but you must then simplify. Multiplying real and imaginary numbers is similar to multiplying with radicals; simply multiply with the imaginary number's coefficient. However, when multiplying imaginary numbers with imaginary numbers, you can eliminate the i, but the resulting number is always negative. Remember that i is the square root of -1, so multiplying i with itself will always yeild a negative number. The same applies to dividing complex numbers.

Powers of i repeat in a pattern:

i¹ = i

i² = 1

i³ = - i

i⁴ = -1

i⁵ = i

...

Negative and Rational Exponents
Exponents, like radicals, have several fundamental properties that are important to memorize: 

• (x^a)(x^b) = x^a+b
• (x^a)^b = x^ab
• x^-a = 1/x^a
• ^a√(x) = x^1/4

When solving an exponential equation, the rules above must be adhered to. However, you may find it necessary to convert exponents into another base (i.e.

4^{2 =} 2⁴). Simply rewrite the given base as an exponent of the desired base, and use the above properties to simplify.