Algebra Notes Flashcards

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Questions and Answers

What do absolute value signs serve as equivalents to?

Braces

Parentheses (correct)

Square brackets

Curly braces

A group of addition and subtraction must be performed from left to right.

True

A group of multiplication and division operations must be performed from right to left.

False

What is a linear equation?

Where all variables have an exponent of 1 Signup and view all the answers

What is the difference between expressions and equations?

Equations contain an equal sign and expressions do not. Signup and view all the answers

What are the steps to simplifying an expression?

<ol> <li>Combine like terms, 2) Find a common denominator, 3) Pull out a common factor, 4) Cancel common factors.</li> </ol> Signup and view all the answers

What are the two ways used to solve systems of equations?

Substitution and Combination/Elimination. Signup and view all the answers

How do you solve an equation using substitution?

<ol> <li>Solve one equation for one variable, 2) Substitute that into the other equation, 3) Solve for the second variable.</li> </ol> Signup and view all the answers

How do you solve an equation using combination/elimination?

Add/Subtract equations to eliminate a variable, then solve for the unknown. Signup and view all the answers

What is key to solving three equations on the GMAT?

Look for symmetries to line up variables without changing the equation. Signup and view all the answers

What is unique about absolute value equations?

False Signup and view all the answers

What is the procedure for solving absolute value equations?

<ol> <li>Isolate the expression within absolute value, 2) Solve for two different cases, 3) Check solutions.</li> </ol> Signup and view all the answers

What is 0 raised to any power?

0 Signup and view all the answers

What is 1 raised to any power?

1 Signup and view all the answers

What are the solutions for x = x²?

Must be either 0 or 1. Signup and view all the answers

What happens to a positive proper fraction when raised to an exponent?

Decreases the value of the expression. Signup and view all the answers

What happens to decimals between 0 and 1 when raised to an exponent?

They decrease. Signup and view all the answers

What is a compound base?

Any base not broken into prime factors or that includes a variable. Signup and view all the answers

How do you multiply same-term bases raised to exponents?

Add the exponents. Signup and view all the answers

How do you divide same-term bases raised to exponents?

Subtract the bottom exponent from the top exponent. Signup and view all the answers

How do you add same-term bases raised to exponents?

Multiply by the base raised to the exponent. Signup and view all the answers

What does an exponent of 0 do?

Anything raised to the power of 0 equals 1. Signup and view all the answers

What happens when a base is raised to a negative exponent?

A negative exponent puts a 1 over the base. Signup and view all the answers

What happens when an exponent is raised to an exponent?

You multiply the exponents. Signup and view all the answers

Study Notes

Absolute Value and Operations

Absolute value signs function like parentheses, indicating the magnitude of a number regardless of its sign.
Addition and subtraction operations are performed in a left-to-right sequence.
Multiplication and division also follow a left-to-right order.

Linear Equations and Expressions

A linear equation features variables to the power of 1.
Equations have an equal sign; expressions do not. Changes in equations affect both sides, while expressions maintain a constant value.

Simplifying Expressions

Combine like terms to simplify.
Identify a common denominator when necessary.
Factor out common elements to simplify further.
Cancel common factors when possible.

Solving Systems of Equations

Systems of equations can be solved through substitution or elimination methods.

Substitution Method

Rearrange one equation to isolate a variable.
Substitute this variable into the other equation.
Solve for the second variable and substitute back to find the first.

Combination/Elimination Method

Align equations by adding or subtracting to eliminate a variable.
Adjust coefficients as needed for ease of elimination.
Solve the resulting equation and substitute back into the original equation for the remaining variable.

Key Strategies for GMAT

Look for symmetries and align variables when solving three equations.

Absolute Value Equations

Typically result in two possible solutions.

Solving Absolute Value Equations

Isolate the expression within absolute value signs.
If |x| = z (z > 0), consider x = ±a, creating two cases.
Validate both solutions by substituting back into the original equation.

Powers of Zero and One

Any number (including 0) raised to any power equals 0.
Any number (including 1) raised to any power equals 1.

Solving for Specific Values

The equation x = x² has solutions of either 0 or 1.
Raising a positive proper fraction to an exponent decreases its value.

Effects of Exponents on Decimals

Decimals between 0 and 1, when raised to an exponent, also decrease in value.

Compound Bases and Exponents

A compound base contains variables or remains undivided into prime factors.
Raising a compound base to an exponent expands each factor’s exponent.

Multiplying and Dividing Bases

To multiply same base terms, add their exponents (e.g., x² × x³ × x⁴ = x⁹).
To divide, subtract the exponent of the denominator from that of the numerator (e.g., 5⁶ ÷ 5² = 5⁴).

Adding and Handling Exponents

Adding same-term bases involves multiplying the count by the base raised to the same exponent (e.g., 3ⁿ + 3ⁿ + 3ⁿ = 3 × 3ⁿ).
An exponent of zero for any base results in 1.
A negative exponent indicates a reciprocal (e.g., 3⁻² = 1/3²).

Exponents Raised to Exponents

When an exponent is raised to another exponent, multiply the exponents (e.g., (z²)⁴ = z⁸).

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