Intermediate Algebra Practice Test 3 – Step-by-Step Study Guidance

Q1. Simplify $\sqrt{12x^2}$

Background

Topic: Simplifying Radical Expressions

This question tests your ability to simplify square roots by factoring and extracting perfect squares.

Key Terms and Formulas:

• Square root ($\sqrt{}$): A value that, when multiplied by itself, gives the original number.

• Perfect square: Numbers like $4, 9, 16, x^2$ whose square roots are integers or monomials.

Step-by-Step Guidance

1. Factor $12x^2$ into $4 \times 3 \times x^2$.

2. Recall that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, so split the square root: $\sqrt{12x^2} = \sqrt{4} \times \sqrt{3} \times \sqrt{x^2}$.

3. Simplify $\sqrt{4}$ and $\sqrt{x^2}$, since both are perfect squares.

Try solving on your own before revealing the answer!

Q2. Simplify $2\sqrt{10}$

Background

Topic: Multiplying Radicals

This question tests your ability to multiply a constant by a radical and recognize if further simplification is possible.

Key Terms and Formulas:

• Radical: An expression containing a root, such as $\sqrt{10}$.

• Multiplication: $a \sqrt{b}$ means $a$ times the square root of $b$.

Step-by-Step Guidance

1. Check if $\sqrt{10}$ can be simplified by factoring $10$into $2 \times 5$ (neither is a perfect square).

2. Multiply the coefficient $2$by the radical$\sqrt{10}$.

Try solving on your own before revealing the answer!

Q3. Approximate $\sqrt{24.415}$ to 3 decimal places

Background

Topic: Estimating Square Roots

This question tests your ability to use a calculator or estimation methods to find the decimal value of a square root.

Key Terms and Formulas:

• Square root approximation: Use a calculator or estimate between known perfect squares.

Step-by-Step Guidance

1. Identify the two perfect squares closest to $24.415$ (e.g., $16$and $25$).

2. Estimate $\sqrt{24.415}$ or use a calculator to find the value to three decimal places.

Try solving on your own before revealing the answer!

Q4. Simplify $x^2 - 5x + 7$

Background

Topic: Simplifying Polynomial Expressions

This question tests your ability to combine like terms and write polynomials in standard form.

Key Terms and Formulas:

• Polynomial: An expression consisting of variables and coefficients.

• Standard form: Arrange terms in descending order of degree.

Step-by-Step Guidance

1. Check for like terms to combine (in this case, none).

2. Write the expression in standard form: $x^2 - 5x + 7$.

Try solving on your own before revealing the answer!

Q5. For $g(x) = \sqrt{x^2 - 5}$, find $g(1)$, $g(5)$, and the domain

Background

Topic: Functions and Domain

This question tests your ability to evaluate a function at specific values and determine the domain based on the expression under the square root.

Key Terms and Formulas:

• Function evaluation: Substitute the given value for $x$ in $g(x)$.

• Domain: The set of all $x$ values for which the function is defined (expression under the square root must be $\geq 0$).

Step-by-Step Guidance

1. Substitute $x = 1$ into $g(x)$: $g(1) = \sqrt{1^2 - 5}$.

2. Substitute $x = 5$ into $g(x)$: $g(5) = \sqrt{5^2 - 5}$.

3. Set $x^2 - 5 \geq 0$ to find the domain.

4. Solve the inequality $x^2 \geq 5$ for $x$.

Try solving on your own before revealing the answer!

Q6. Simplify $\sqrt{50}$

Background

Topic: Simplifying Radicals

This question tests your ability to factor the radicand and extract perfect squares.

Key Terms and Formulas:

• Radicand: The number inside the square root.

• Perfect square factors: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.

Step-by-Step Guidance

1. Factor $50$into $25 \times 2$.

2. Split the square root: $\sqrt{50} = \sqrt{25} \times \sqrt{2}$.

3. Simplify $\sqrt{25}$.

Try solving on your own before revealing the answer!

Q7. Rationalize the denominator: $\frac{7b}{\sqrt{6}}$

Background

Topic: Rationalizing Denominators

This question tests your ability to eliminate radicals from the denominator by multiplying by a suitable form of 1.

Key Terms and Formulas:

• Rationalizing: Multiply numerator and denominator by $\sqrt{6}$.

• $\frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}$

Step-by-Step Guidance

1. Multiply numerator and denominator by $\sqrt{6}$.

2. Write the new numerator: $7b \times \sqrt{6}$.

3. Write the new denominator: $\sqrt{6} \times \sqrt{6}$.

Try solving on your own before revealing the answer!

Q8. Rationalize the denominator: $\frac{\sqrt{50}}{3}$

Background

Topic: Rationalizing Denominators

This question tests your ability to simplify radicals and ensure the denominator is rational.

Key Terms and Formulas:

• Rationalizing: If the denominator is already rational, focus on simplifying the numerator.

• Simplify $\sqrt{50}$ as in Q6.

Step-by-Step Guidance

1. Simplify $\sqrt{50}$ as $\sqrt{25 \times 2}$.

2. Write the fraction with the simplified numerator.

Try solving on your own before revealing the answer!

Q9. Add: $\frac{9}{5} + \frac{8}{5}$

Background

Topic: Adding Fractions

This question tests your ability to add fractions with common denominators.

Key Terms and Formulas:

• Common denominator: When denominators are the same, add numerators.

• $\frac{a}{d} + \frac{b}{d} = \frac{a+b}{d}$

Step-by-Step Guidance

1. Add the numerators: $9 + 8$.

2. Keep the denominator the same: $5$.

Try solving on your own before revealing the answer!

Q10. Simplify $\frac{113\sqrt{7}}{72}$

Background

Topic: Simplifying Radical Fractions

This question tests your ability to simplify fractions with radicals in the numerator.

Key Terms and Formulas:

• Fraction simplification: Reduce numerator and denominator if possible.

• Radical: $\sqrt{7}$ remains as is unless $7$ is a perfect square.

Step-by-Step Guidance

1. Check if $113$and $72$ have common factors to reduce.

2. Write the fraction in simplest form.

Try solving on your own before revealing the answer!

Final Answer Examples

Q1: $2x\sqrt{3}$ Q2: $2\sqrt{10}$ Q3: $4.941$ Q4: $x^2 - 5x + 7$ Q5: $g(1)$ is not defined, $g(5) = 4$, domain: $x \leq -\sqrt{5}$ or $x \geq \sqrt{5}$ Q6: $5\sqrt{2}$ Q7: $\frac{7b\sqrt{6}}{6}$ Q8: $\frac{5\sqrt{2}}{3}$ Q9: $\frac{17}{5}$ Q10: $\frac{113\sqrt{7}}{72}$

Each answer is obtained by following the step-by-step process outlined above.