The Order of Operations for Simplifying Algebraic Expressions:
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.
X^( Y / Z ) = ( The Z-th Root of X )^Y = The Z-th Root of ( X^Y )
Multiplying out a Factored Expression:
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.
- 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.