TEAS Mathpart2

High Yield Topics for Exam Review

The focus of this guide is on high-yield topics guaranteed to appear on the exam.
Key areas include the following:

• Operations

• Exponents and Roots

• Rational Numbers (Fractions and Decimals)

Operations

Key operations that will almost certainly appear on the exam are as follows:

• Addition

• Subtraction

• Multiplication

• Division

Order of Operations

• The order in which operations are performed in multi-step expressions is governed by the acronym PEMDAS:

  • P: Parentheses

  • E: Exponents

  • M: Multiplication

  • D: Division

  • A: Addition

  • S: Subtraction

• PEMDAS Breakdown:

  • Always resolve operations inside parentheses first.

  • For multiplication and division, perform from left to right as they have equal precedence.

  • The same applies to addition and subtraction.

Rules for Operations

• Signs in Multiplication/Division:

  • Multiplying or dividing a positive number maintains its sign.

  • Multiplying or dividing a negative changes the sign.

  • Double Negatives stay the same (e.g., -(-2) = 2).

Exponents and Roots

Understanding Exponents

• An exponent indicates how many times the base is multiplied by itself.

• Examples:

  • $a^1 = a$

  • $a^0 = 1$ (for any non-zero number a)

• Exponent Rules:

  • Same Base Multiplication:

  • $a^m imes a^n = a^{m+n}$

  • Same Base Division:

  • $rac{a^m}{a^n} = a^{m-n}$

  • Power of a Power:

  • $(a^m)^n = a^{m imes n}$

  • Negative Exponents:

  • $a^{-n} = rac{1}{a^n}$

• To remove the exponent, use the square root for powers of two.

  • Example: $a^2 = x <br>ightarrow a = ext{sqrt}(x)$

Examples of Exponent Rules

1. Same Base Multiplication:

   • If $10^2 imes 10^4$, then:

     • Exponents are added: $2 + 4 = 6$

     • Hence, $10^6$.

2. Same Base Division:

   • If $rac{10^4}{10^2}$, then:

     • Exponents are subtracted: $4 - 2 = 2$

     • Hence, $10^2$.

3. Power of a Power:

   • If $(10^4)^2$, then:

     • Multiply exponents: $4 imes 2 = 8$

     • Hence, $10^8$.

4. Negative Exponent:

   • If $10^{-2}$, represents:

     • $rac{1}{10^2} = rac{1}{100}$.

Rational Numbers

Definition and Types of Rational Numbers

• Rational Numbers: numbers that can be expressed as a fraction $rac{a}{b}$, including integers.

  • Examples:

  • The number 3 can be written as $rac{3}{1}$.

  • The number 4 can also be written as $rac{4}{1}$.

Fractions: Definition and Types

• Proper Fractions:

  • The numerator (top number) is less than the denominator (bottom number). Example: $rac{1}{2}$.

• Improper Fractions:

  • The numerator is greater than or equal to the denominator. Example: $rac{5}{4}$.

• Mixed Numbers:

  • Combination of a whole number and a proper fraction, like $2 rac{3}{4}$. Convert to an improper fraction:

  • $2 rac{3}{4} = rac{(2 imes 4) + 3}{4} = rac{8 + 3}{4} = rac{11}{4}$.

Operations on Fractions

Addition and Subtraction

• To add or subtract fractions, find a common denominator.

  • Example: $rac{1}{2} + rac{2}{4}$

  • Convert $rac{1}{2}$ to $rac{2}{4}$ for easier addition:

  • Therefore, $rac{2}{4} + rac{2}{4} = rac{4}{4} = 1$.

Multiplication and Division

• To multiply fractions, multiply straight across:

  • $rac{1}{2} imes rac{2}{4} = rac{1 imes 2}{2 imes 4} = rac{2}{8} = rac{1}{4}$.

• To divide fractions, multiply by the reciprocal of the second fraction:

  • $rac{1}{2} ext{ divided by } rac{2}{4} ext{ becomes } rac{1}{2} imes rac{4}{2} = rac{4}{4} = 1$.

Percentages

• Conversion Methods:

  • To convert from a percentage to a decimal, move the decimal point two places to the left or divide by 100.

  • Conversely, to convert a decimal to a percentage, move the decimal two places to the right or multiply by 100.

  • Example:

    • 100% = 1.0 as a decimal

    • and 0.45 becomes 45%.

Practical Problem Solving

Example Problems that used PEMDAS and Exponents:

1. For the expression $5 ext{ divided by } (2 + 3)^2 - 6$:

   • Start with Parentheses: $5 ext{ divided by } (5)^2 - 6$

   • Next Exponent: $5 ext{ divided by } 25 - 6 = rac{5}{25} - 6$

   • Perform Division and Subtraction following PEMDAS.

   • Final answer is calculated based on the processed order following PEMDAS.

2. Simplifying $2^{-3}$ using negative exponents:

   • $2^{-3} = rac{1}{2^{3}} = rac{1}{2 imes 2 imes 2} = rac{1}{8}$.

3. Dividing $rac{3}{4}$ by $rac{1}{2}$:

   • The reciprocal of $rac{1}{2}$ is $2$, thus $rac{3}{4} imes 2 = rac{6}{4}$; simplified to $1.5$ or $1 rac{1}{2}$.

4. Mixed number times whole number example:

   • $1 rac{1}{2} imes 6$ becomes $rac{3}{2} imes rac{6}{1} = rac{18}{2} = 9$.

Conclusion:

• Familiarize yourself with these operations and rules, and practice more examples to build confidence.

• Always keep the order of operations (PEMDAS) in mind when solving problems and simplifying expressions.