The Order of Operations for Simplifying Algebraic Expressions:

PEMDAS:

1. Parentheses

2. Exponents

3. Multiplication

3. Division

4. Addition

4. Subtraction

When two or more operations are at the same level of priority, always work from left to right.

Distributing an Exponent to a Fraction:

( A / B )^C = A^C / B^C

Distributing an Exponent to a Product:

C = A * B

C^X = ( A * B )^X = A^X * B^X

Multiplying Exponential Terms with Common Bases:

X^A * X^B = X^( A + B )

Dividing Exponential Terms with Common Bases:

X^A / X^B = X^( A - B )

Raising a Number by an Exponent of 0:

X^0 = 1

Raising a Number by a Negative Exponent:

X^( -A ) = 1 / X^A

How to Handle Nested Exponents?

( X^A )^B = X^( A * B )

Raising a Fraction by a Negative Exponent:

( A / B ) ^( -C ) = ( B / A )^C

Factoring out a Common Term:

X^A + X^( A + 1 ) = X^A * ( X^0 + X^1 ) = X^A * ( 1 + X )

The Square Root of a Variable:

If X = Square Root of 16, then X = +4.

The Square Root of a Squared Variable:

If X^2 = 16, then X_1 = +4 and X_2 = -4.

Fractional Exponents:

X^( Y / Z ) = ( The Z-th Root of X )^Y = The Z-th Root of ( X^Y )

Multiplying out a Factored Expression:

FOIL:

1. First

2. Outer

3. Inner

4. Last

The Three Special Products:

1. ( X + Y )^2 = X^2 + 2 * X * Y + Y^2

2. ( X - Y )^2 = X^2 - 2 * X * Y + Y^2

3. ( X + Y ) * ( X - Y ) = X^2 - Y^2

The Three Most Common Inequality Statements:

1. X * Y > 0 means that X and Y are both positive or both negative.

2. X * Y < 0 means that X and Y have different signs.

3. X^2 - X < 0 means that X^2 < X which in turn means that 0 < X < 1.

How to Simplify a Fraction with a Simple Square Root in the Denominator?

You just have to multiply the numerator and the denominator by the square root.

How to Simplify a Fraction with a Denominator that Contains the Sum or Difference of a Square Root and Another Term?

Here, you have to multiply the numerator and the denominator with the conjugate of the denominator.

For ( A + Square Root of B ), the conjugate is given by ( A - Square Root of B ), and vice versa.

The Reciprocals of Inequalities:

- If X < Y, then ( 1 / X ) > ( 1 / Y ) when X and Y are positive.

- If X < Y, then ( 1 / X ) > ( 1 / Y ) when X and Y are negative.

- If X < Y, then ( 1 / X ) < ( 1 / Y ) when X is negative and Y is positive.

Squaring Inequalities:

- If both sides are known to be negative, then flip the inequality sign when you square.

- If both sides are known to be positive, then do not flip the inequality sign when you square.

- If one side is positive and the other one is negative, then you cannot square at all.

- If one or both signs are unclear, then you also cannot square.

How to Determine Roots?

In order to determine the root of a number, break the number into its prime factors.