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Flashcards in Algebra Deck (21):
1

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.

2

Distributing an Exponent to a Fraction:

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

3

Distributing an Exponent to a Product:

C = A * B

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

4

Multiplying Exponential Terms with Common Bases:

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

5

Dividing Exponential Terms with Common Bases:

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

6

Raising a Number by an Exponent of 0:

X^0 = 1

7

Raising a Number by a Negative Exponent:

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

8

How to Handle Nested Exponents?

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

9

Raising a Fraction by a Negative Exponent:

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

10

Factoring out a Common Term:

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

11

The Square Root of a Variable:

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

12

The Square Root of a Squared Variable:

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

13

Fractional Exponents:

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

14

Multiplying out a Factored Expression:

FOIL:

1. First

2. Outer

3. Inner

4. Last

15

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

16

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.

17

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.

18

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.

19

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.

20

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.

21

How to Determine Roots?

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