Comprehensive Step-by-Step Guidance for Beginning-Intermediate Algebra Review

Q1. Is 182 a prime or composite number?

Background

Topic: Prime and Composite Numbers

This question tests your understanding of the definitions of prime and composite numbers and your ability to determine which category a given number falls into.

Key Terms:

• Prime Number: A number greater than 1 that has only two positive divisors: 1 and itself.

• Composite Number: A number greater than 1 that has more than two positive divisors.

Step-by-Step Guidance

1. Check if 182 is greater than 1 (since only numbers greater than 1 can be prime or composite).

2. Test if 182 can be divided evenly by any integer other than 1 and itself (for example, try dividing by 2, 3, 5, etc.).

3. If you find a divisor other than 1 and 182, the number is composite; otherwise, it is prime.

Try solving on your own before revealing the answer!

Q2. Find the prime factorization of 294.

Background

Topic: Prime Factorization

This question tests your ability to break down a composite number into a product of prime numbers.

Key Terms and Process:

• Prime Factorization: Expressing a number as a product of its prime factors.

Step-by-Step Guidance

1. Start by dividing 294 by the smallest prime number (2) and continue dividing by prime numbers (3, 5, 7, etc.) until you reach only prime numbers.

2. Write each division as an equation, showing the quotient and the divisor.

3. Continue factoring each quotient until all factors are prime.

4. Express 294 as a product of these prime numbers.

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Q3. Find the least common multiple (LCM) of 8, 18, and 24.

Background

Topic: Least Common Multiple (LCM)

This question tests your ability to find the smallest number that is a multiple of all the given numbers.

Key Terms and Formula:

• LCM: The smallest positive integer that is a multiple of two or more numbers.

• Prime factorize each number to help find the LCM.

Step-by-Step Guidance

1. Prime factorize each number: 8, 18, and 24.

2. For each prime factor, take the highest power that appears in any of the numbers.

3. Multiply these highest powers together to get the LCM.

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Q4. Write the fraction 5/6 with a denominator of 18.

Background

Topic: Equivalent Fractions

This question tests your ability to write an equivalent fraction with a specified denominator.

Key Terms and Formula:

• Equivalent Fractions: Fractions that represent the same value, even though they may have different numerators and denominators.

• To find an equivalent fraction, multiply the numerator and denominator by the same number.

Step-by-Step Guidance

1. Determine what number you need to multiply 6 by to get 18.

2. Multiply both the numerator (5) and the denominator (6) by this number.

3. Write the new fraction with denominator 18.

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Q5. Reduce the fraction 10/14, if possible.

Background

Topic: Simplifying Fractions

This question tests your ability to reduce a fraction to lowest terms by dividing both numerator and denominator by their greatest common factor (GCF).

Key Terms and Process:

• Greatest Common Factor (GCF): The largest integer that divides both the numerator and denominator.

Step-by-Step Guidance

1. Find the GCF of 10 and 14.

2. Divide both the numerator and denominator by the GCF.

3. Write the simplified fraction.

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Q6. Perform the operation: 12 + (−4) − 3 + 8. Write in lowest terms.

Background

Topic: Integer Operations

This question tests your ability to add and subtract integers, following the correct order of operations.

Key Concepts:

• Addition and subtraction of positive and negative numbers.

Step-by-Step Guidance

1. Start by evaluating the expression inside the parentheses: (−4).

2. Add 12 and (−4).

3. Subtract 3 from the result.

4. Add 8 to the new result.

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Q7. Evaluate (4)(−6)(−3).

Background

Topic: Multiplying Integers

This question tests your understanding of multiplying positive and negative numbers.

Key Concepts:

• Multiplying two negative numbers results in a positive product.

• Multiplying a positive and a negative number results in a negative product.

Step-by-Step Guidance

1. Multiply the first two numbers: 4 × (−6).

2. Take the result and multiply by (−3).

3. Pay attention to the sign changes at each step.

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Q8. Evaluate the expression: |−183|.

Background

Topic: Absolute Value

This question tests your understanding of the absolute value function, which measures the distance from zero on the number line.

Key Terms:

• Absolute Value: $|a|$ is always non-negative, regardless of whether $a$ is positive or negative.

Step-by-Step Guidance

1. Recognize that $|−183|$ means the distance from 0 to −183.

2. Recall that the absolute value of a negative number is its positive counterpart.

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Q9. Evaluate the square root: $\sqrt{36}$

Background

Topic: Square Roots

This question tests your ability to find the principal (positive) square root of a perfect square.

Key Terms:

• Square Root: $\sqrt{a}$ is the number that, when multiplied by itself, gives $a$.

Step-by-Step Guidance

1. Identify the number whose square is 36.

2. Recall that the square root of a positive perfect square is a positive integer.

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Q10. Simplify the expression: $\sqrt{16} + \sqrt{100}$

Background

Topic: Simplifying Radicals

This question tests your ability to evaluate and add square roots of perfect squares.

Key Terms:

• Square Root: $\sqrt{a}$

Step-by-Step Guidance

1. Evaluate $\sqrt{16}$.

2. Evaluate $\sqrt{100}$.

3. Add the two results together.

Try solving on your own before revealing the answer!