Precalculus Test 1 Review: Step-by-Step Study Guidance

Q1. Convert the following radicals to rational exponent notation.

• (a) $\sqrt[3]{9x-1}$

• (b) $\sqrt[4]{4x^2-2}$

• (c) $\sqrt{5x-3}$

Background

Topic: Exponents and Radicals

This question tests your understanding of how to rewrite radical expressions using rational (fractional) exponents.

Key Terms and Formulas

• Radical notation: $\sqrt[n]{a}$

• Rational exponent notation: $a^{1/n}$

• General rule: $\sqrt[n]{a^m} = a^{m/n}$

Step-by-Step Guidance

1. Identify the index of the radical (the small number outside the radical sign). If there is no number, it is a square root (index 2).

2. Rewrite the expression inside the radical as the base.

3. Express the radical as an exponent with a fractional power: the numerator is the exponent of the base (if any), and the denominator is the index of the root.

4. For each part, set up the rational exponent form, but do not simplify further yet.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $(9x-1)^{1/3}$

• (b) $(4x^2-2)^{1/4}$

• (c) $(5x-3)^{1/2}$

Each radical is rewritten as a power with a fractional exponent, where the denominator is the root and the numerator is the power.

Q2. Convert the following to radical notation.

• (a) $x^{3/5}$

• (b) $(2x+3)^{5/4}$

• (c) $(3x-5)^{7/9}$

Background

Topic: Exponents and Radicals

This question checks your ability to convert expressions with rational exponents back into radical notation.

Key Terms and Formulas

• Rational exponent: $a^{m/n}$

• Radical notation: $\sqrt[n]{a^m}$

• General rule: $a^{m/n} = \sqrt[n]{a^m}$ or $\left(\sqrt[n]{a}\right)^m$

Step-by-Step Guidance

1. Identify the base, numerator (power), and denominator (root) in the exponent.

2. Rewrite the expression as a radical, using the denominator as the index of the root and the numerator as the power.

3. Set up the radical notation for each part, but do not simplify further yet.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $\sqrt[5]{x^3}$

• (b) $\sqrt[4]{(2x+3)^5}$

• (c) $\sqrt[9]{(3x-5)^7}$

Each rational exponent is converted to a radical, with the denominator as the root and the numerator as the power.

Q3. Rationalize the denominator and simplify completely.

• (a) $\frac{8}{3\sqrt{6}}$

• (b) $\frac{x^{3/2}}{\sqrt{x+4}}$

Background

Topic: Rationalizing Denominators

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

Key Terms and Formulas

• Rationalizing the denominator: Multiplying numerator and denominator by a value that will remove the radical from the denominator.

• For $\sqrt{a}$ in the denominator, multiply by $\sqrt{a}$.

• For $a + \sqrt{b}$, multiply by the conjugate $a - \sqrt{b}$.

Step-by-Step Guidance

1. Identify the radical in the denominator.

2. Multiply numerator and denominator by the appropriate radical to eliminate the root in the denominator.

3. Simplify the resulting numerator and denominator as much as possible.

4. For expressions with exponents, use exponent rules to combine or simplify terms.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $\frac{8}{3\sqrt{6}} = \frac{8\sqrt{6}}{18} = \frac{4\sqrt{6}}{9}$

• (b) $\frac{x^{3/2}}{\sqrt{x+4}} = \sqrt{x}(x+4)$

Multiplying by the appropriate radical removes the root from the denominator and allows for further simplification.

Q4. Rationalize the numerator and simplify completely.

• (a) $\frac{\sqrt{x}-\sqrt{y}}{x-y}$

• (b) $\frac{\sqrt{x}-a}{x-a^2}$

Background

Topic: Rationalizing Numerators

This question tests your ability to eliminate radicals from the numerator by multiplying by the conjugate.

Key Terms and Formulas

• Conjugate: For $a - b$, the conjugate is $a + b$.

• Multiplying by the conjugate: $(a-b)(a+b) = a^2 - b^2$

Step-by-Step Guidance

1. Identify the radical in the numerator.

2. Multiply numerator and denominator by the conjugate of the numerator.

3. Expand the numerator using the difference of squares formula.

4. Simplify the denominator as needed, but stop before the final simplification.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $\frac{\sqrt{x}-\sqrt{y}}{x-y} \cdot \frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}} = \frac{x-y}{(x-y)(\sqrt{x}+\sqrt{y})} = \frac{1}{\sqrt{x}+\sqrt{y}}$

• (b) $\frac{\sqrt{x}-a}{x-a^2} \cdot \frac{\sqrt{x}+a}{\sqrt{x}+a} = \frac{x-a^2}{(x-a^2)(\sqrt{x}+a)} = \frac{1}{\sqrt{x}+a}$

Multiplying by the conjugate eliminates the radical in the numerator and allows for further simplification.

Q5. Solve the following linear equations.

• (a) $-2(6-x) = 12(x+1)$

• (b) $\frac{3x+5}{7} + 2 = \frac{5x}{3}$

• (c) $7x-14 = -4(3-2x)$

Background

Topic: Linear Equations

This question tests your ability to solve equations involving variables on both sides, including those with fractions.

Key Terms and Formulas

• Linear equation: An equation of the form $ax + b = cx + d$

• To solve: Isolate $x$ by performing inverse operations.

Step-by-Step Guidance

1. Expand any parentheses and distribute as needed.

2. Combine like terms on each side of the equation.

3. Move all terms containing $x$ to one side and constants to the other.

4. For equations with fractions, multiply both sides by the least common denominator to clear fractions.

5. Stop before the final calculation for $x$.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $x = -\frac{3}{2}$

• (b) $x = \frac{11}{4}$

• (c) $x = -2$

Each equation is solved by isolating $x$ using algebraic operations.

Q6. Solve the following linear inequalities. Write your answers in interval notation.

• (a) $-11x + 3(3x-1) \geq 5$

• (b) $2 < \frac{5x-1}{3} \leq 4$

• (c) $-4 < 1-5x < 11$

Background

Topic: Linear Inequalities

This question tests your ability to solve inequalities and express the solution in interval notation.

Key Terms and Formulas

• Linear inequality: An inequality involving a linear expression.

• Interval notation: A way to write the set of solutions using parentheses and brackets.

• Remember: When multiplying or dividing both sides by a negative, reverse the inequality sign.

Step-by-Step Guidance

1. Expand and simplify each side of the inequality.

2. Isolate $x$ by performing inverse operations.

3. For compound inequalities, solve each part separately and find the intersection.

4. Express the solution in interval notation, but stop before writing the final interval.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $(-\infty, -4]$

• (b) $(19, 33]$

• (c) $(-2, 1)$

Each inequality is solved for $x$ and the solution is written in interval notation.

Q7. Factor the following polynomials completely.

• (a) $-15x^3 - 30x^2 + 45x$

• (b) $121x^2 - 36$

• (c) $x^2 - 11x + 28$

• (d) $4x^2 + 7x - 2$

Background

Topic: Factoring Polynomials

This question tests your ability to factor polynomials using various methods, such as factoring out the greatest common factor, difference of squares, and factoring trinomials.

Key Terms and Formulas

• Greatest common factor (GCF): The largest factor shared by all terms.

• Difference of squares: $a^2 - b^2 = (a-b)(a+b)$

• Factoring trinomials: $ax^2 + bx + c$

Step-by-Step Guidance

1. Look for a GCF and factor it out if possible.

2. For quadratics, look for patterns such as difference of squares or trinomials that can be factored into two binomials.

3. Set up the factors, but stop before writing the fully factored form.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $-15x(x^2 + 2x - 3) = -15x(x+3)(x-1)$

• (b) $(11x-6)(11x+6)$

• (c) $(x-4)(x-7)$

• (d) $(4x-1)(x+2)$

Each polynomial is factored completely using appropriate methods.

Q8. Solve the following equations.

• (a) $x^2 = -4x + 21$

• (b) $x^2 - 7x - 18 = 0$

• (c) $3x^2 + 8x - 3 = 0$

• (d) $5x^2 + 4x = 12$

• (e) $(x-5)^2 - 1 = 80$

• (f) $-6(x+5)^2 = -600$

• (g) $x^3 - 10x^2 + 16x = 0$

• (h) $x^3 - 48x^2 + 47x = 0$

• (i) $x^4 - 6x^2 + 5 = 0$

• (j) $x^4 - 8x^2 + 12 = 0$

Background

Topic: Solving Quadratic and Higher-Degree Equations

This question tests your ability to solve quadratic and cubic equations by factoring, using the quadratic formula, or other algebraic methods.

Key Terms and Formulas

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Expressing the equation as a product of factors and setting each factor to zero.

• Zero product property: If $ab=0$, then $a=0$ or $b=0$.

Step-by-Step Guidance

1. Rewrite the equation in standard form (all terms on one side, set equal to zero).

2. Identify if the equation can be factored or if the quadratic formula is needed.

3. Factor the equation or set up the quadratic formula as appropriate.

4. Stop before calculating the final values for $x$.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $x = -7, 3$

• (b) $x = 9, -2$

• (c) $x = -3, \frac{1}{3}$

• (d) $x = -2, \frac{6}{5}$

• (e) $x = 14, -4$

• (f) $x = -15, 5$

• (g) $x = 0, 2, 8$

• (h) $x = 0, 1, 47$

• (i) $x = -1, 1, -\sqrt{5}, \sqrt{5}$

• (j) $x = -\sqrt{2}, \sqrt{2}, -\sqrt{6}, \sqrt{6}$

Each equation is solved using factoring or the quadratic formula as appropriate.

Q9. Solve the following radical equations.

• (a) $\sqrt{10-5x} = x+8$

• (b) $\sqrt{2x+33} = x-1$

Background

Topic: Radical Equations

This question tests your ability to solve equations involving square roots. Remember to check for extraneous solutions.

Key Terms and Formulas

• Radical equation: An equation in which the variable is inside a radical.

• To solve: Isolate the radical and square both sides to eliminate the root.

• Check solutions: Squaring both sides can introduce extraneous solutions.

Step-by-Step Guidance

1. Isolate the radical on one side of the equation.

2. Square both sides to eliminate the square root.

3. Solve the resulting equation for $x$.

4. Check all solutions in the original equation to ensure they are valid.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $x = -3$

• (b) $x = 8$

Always check your solutions in the original equation to avoid extraneous answers.

Q10. Solve the following equations by completing the square.

• (a) $x^2 - 14x - 2 = 0$

• (b) $x^2 + 16x + 39 = 0$

Background

Topic: Completing the Square

This question tests your ability to solve quadratic equations by rewriting them in a perfect square form.

Key Terms and Formulas

• Completing the square: Adding and subtracting the same value to create a perfect square trinomial.

• General form: $x^2 + bx + c = 0$

• Perfect square trinomial: $x^2 + 2hx + h^2 = (x+h)^2$

Step-by-Step Guidance

1. Move the constant term to the other side of the equation.

2. Find the value to complete the square: $(\frac{b}{2})^2$.

3. Add this value to both sides of the equation.

4. Rewrite the left side as a squared binomial, but stop before solving for $x$.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $x = 7 + \sqrt{51},\ 7 - \sqrt{51}$

• (b) $x = -13,\ -3$

Completing the square allows you to solve quadratics that may not factor easily.

Q11. Solve the following equations by using the Quadratic Formula (simplify completely).

• (a) $2x^2 - 5x - 9 = 0$

• (b) $x^2 + 4x = 3$

Background

Topic: Quadratic Formula

This question tests your ability to apply the quadratic formula to solve quadratic equations.

Key Terms and Formulas

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Standard form: $ax^2 + bx + c = 0$

Step-by-Step Guidance

1. Write the equation in standard form.

2. Identify $a$, $b$, and $c$.

3. Plug these values into the quadratic formula.

4. Simplify under the square root, but stop before calculating the final values for $x$.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $x = \frac{5 + \sqrt{97}}{4},\ \frac{5 - \sqrt{97}}{4}$

• (b) $x = -2 + \sqrt{7},\ -2 - \sqrt{7}$

The quadratic formula provides solutions for any quadratic equation, even when factoring is not possible.

Q12. Perform the indicated operation.

• (a) $\frac{3x}{x^2-9} - \frac{2}{x+3}$

• (b) $\frac{x^2-7x+10}{x^2-4x+3} - \frac{x^2-9}{x^2-4x+3}$

• (c) $\frac{x^2+3x}{x^2-4} + \frac{2x}{x^2-4}$

Background

Topic: Operations with Rational Expressions

This question tests your ability to add and subtract rational expressions by finding common denominators and combining numerators.

Key Terms and Formulas

• Rational expression: A fraction with polynomials in the numerator and denominator.

• Common denominator: The least common multiple of the denominators.

• Combine numerators over the common denominator.

Step-by-Step Guidance

1. Factor denominators where possible.

2. Find the least common denominator (LCD).

3. Rewrite each fraction with the LCD as the denominator.

4. Combine the numerators and simplify, but stop before the final simplification.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $\frac{x(x-6)}{(x-3)(x+3)}$

• (b) $\frac{(x-5)(x+2)}{(x-1)(x-3)}$

• (c) $\frac{(x+2)(x+3)}{x^2-4}$

Combining rational expressions requires a common denominator and careful simplification.

Q13. Simplify the following rational expressions completely and state the values of x that must be excluded.

• (a) $\frac{x^2-49}{x^2-7x}$

• (b) $\frac{x^2-8x+15}{x^2+3x-18}$

• (c) $\frac{x^2+5x-6}{x^2+5x+6}$

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor and simplify rational expressions, and to identify excluded values (values that make the denominator zero).

Key Terms and Formulas

• Excluded values: Values of $x$ that make the denominator zero.

• Factor numerator and denominator, then cancel common factors.

Step-by-Step Guidance

1. Factor the numerator and denominator completely.

2. Identify and cancel any common factors.

3. List all values of $x$ that make the denominator zero (excluded values).

4. Stop before writing the final simplified expression and excluded values.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $\frac{x+7}{x}$; $x \neq 0, 7$

• (b) $\frac{x-5}{x-3}$; $x \neq 8, -3$

• (c) $\frac{x-1}{x+6}$; $x \neq -6, 1$

Always state excluded values after simplifying rational expressions.

Q14. Simplify the following complex rational expressions.

• (a) $\frac{2}{x} \div \frac{1}{x^2}$

• (b) $\frac{3x-4}{x+4} \div \frac{x-3}{x}$

• (c) $\frac{x^2+2x}{x^2-4}$

Background

Topic: Complex Rational Expressions

This question tests your ability to simplify expressions that involve division of rational expressions.

Key Terms and Formulas

• To divide by a fraction, multiply by its reciprocal.

• Factor numerators and denominators where possible.

Step-by-Step Guidance

1. Rewrite the division as multiplication by the reciprocal.

2. Factor all numerators and denominators.

3. Multiply across numerators and denominators.

4. Simplify by canceling common factors, but stop before the final simplification.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $2x$

• (b) $\frac{(3x-4)x}{(x+4)(x-3)}$

• (c) $\frac{x(x+2)}{(x+2)(x-2)} = \frac{x}{x-2}$

Multiplying by the reciprocal and simplifying gives the final result.