A Concise Workbook for College Algebra

Lesson 14: Radical Expressions: Concepts and Properties

Radical Expressions

• Definition: If $b^2 = a$, then $b$ is called a square root of $a$.

  • Notation: The positive square root of $a$ is denoted by $\sqrt{a}$, referred to as the principal square root.

• Function Definition: The function $f(x) = \sqrt{x}$ is termed a square root function.

  • Domain: As a real-valued function, the domain consists of all real numbers $x$ such that $x ≥ 0$, which can be expressed in interval notation as $[0, +\infty)$.

• Simplification: For any real number $a$, the expression $\sqrt{a^2}$ can be simplified to $\sqrt{a^2} = |a|$.

Cube Roots

• Definition: If $b^3 = a$, then $b$ is classified as a cube root of $a$.

  • Notation: The cube root of a real number $a$ is denoted as $\sqrt[3]{a}$.

• Simplification: For any real number $a$, the expression $\sqrt[3]{a^3}$ simplifies to $\sqrt[3]{a^3} = a$.

N-th Roots

• General Definition: If $b^n = a$, then $b$ is termed an n-th root of $a$.

  • Notation: If $n$ is even, the positive n-th root is referred to as the principal n-th root, denoted by $\sqrt[n]{a}$.

  • If $n$ is odd, then the n-th root $\sqrt[n]{a}$ maintains the same sign as $a$.

• Components:

  • Radical Sign: In $\sqrt[n]{a}$, the symbol $\sqrt{}$ is the radical sign.

  • Radicand: In $\sqrt[n]{a}$, $a$ is the radicand.

  • Index: In $\sqrt[n]{a}$, $n$ is referred to as the index.

• Restrictions: If $n$ is even, the n-th root of a negative number is not a real number.

• Simplification Rules:

  1. $\sqrt[n]{a^n} = |a|$ if $n$ is even.

  2. $\sqrt[n]{a^n} = a$ if $n$ is odd.

• Radical simplification means that the radicand has no perfect power factors that are applicable against the radical.

Examples of Radical Simplification

• Example 14.1: Simplify the radical expression using the definition.

  1. $\sqrt[4]{(y-1)^2}$

  • Solution: $\sqrt[4]{(y-1)^2} = \sqrt{[2(y-1)]^2} = 2|y-1|$.

  1. $\sqrt[3]{-8x^3y^6}$

  • Solution: $\sqrt[3]{(-2xy^2)^2} = -2xy^2$.

Rational Exponents

• Definition: If $\sqrt[n]{a}$ is a real number, then we define a rational exponent as follows:

  • $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

Properties of Rational Exponents

1. $a^m \cdot a^n = a^{m+n}$.

2. $\frac{a^m}{a^n} = a^{m-n}$.

3. $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$.

4. $(a^m)^n = a^{mn}$.

5. $(ab)^m = a^m \cdot b^m$.

6. $\sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}}$.

Examples of Simplifying Expressions with Rational Exponents

• Example 14.2:

  1. $\sqrt{x}\sqrt[3]{x^2}$

  • Solution: $\sqrt{x}\sqrt[3]{x^2} = x^{\frac{1}{2}} \cdot x^{\frac{2}{3}} = x^{\frac{1}{2}+\frac{2}{3}} = x^{\frac{7}{6}} = x\sqrt[6]{x}$.

  1. $\sqrt[3]{\sqrt{x^3}}$

  • Solution: $\sqrt[3]{(x^3)^{\frac{1}{2}}} = (x^{3})^{\frac{1}{6}} = x^{\frac{3}{6}} = x^{\frac{1}{2}} = \sqrt{x}$.

Exercises for Practice

Exercise 14.1: Evaluate the square root. If not a real number, state so.

1. $−\sqrt{\frac{4}{25}}$ (not a real number)

2. $\sqrt{49}-\sqrt{9}$

3. $−\sqrt{−1}$

Exercise 14.2: Simplify the radical expressions.

Exercise 14.3: Simplify the radical expressions.

Exercise 14.4: Simplify the radical expression, assuming all variables are positive.

Lesson 15: Algebra of Radicals

Product and Quotient Rules for Radicals

• If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, then the multiplication is as follows:
  $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.

• If $b \neq 0$, $\sqrt[n]{a} \div \sqrt[n]{b} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$.

Examples

1. Example 15.1: Simplify: $\sqrt[4]{8xy^4} \cdot \sqrt[4]{2x^7y}$.

   • Solution: $\sqrt[4]{8xy^4 \cdot 2x^7y} = \sqrt[4]{16x^8y^5}$.

2. Example 15.2: Combine like radicals. $\sqrt{8x^3} − \sqrt{(-2)^2x^4} + \sqrt{2x^5}$:

   • Solution: $2x\sqrt{2x} − 2x^2 + x^2\sqrt{2x}$.

Rationalizing Denominators

• Definition: Rationalizing the denominator means rewriting a radical expression into an equivalent expression where the denominator has no radicals.

• Example 15.4:

  1. $\frac{1}{2\sqrt{x^3}}$

  • Multiply the expression by $\frac{\sqrt{x}}{\sqrt{x}}$.

  1. $\frac{\sqrt{x} \cdot 1}{2\sqrt{x^2}}$.

Exercises

Exercise 15.1 through Exercise 15.9: Various exercises on simplifying, adding, subtracting, and rationalizing radical expressions.

Lesson 16: Solve Radical Equations

Techniques for Solving Radical Equations

• The process generally involves isolating the radical expression and taking the appropriate power of both sides to eliminate the radical.

Example Problems

• Example 16.1: Solve $x - \sqrt{x + 1} = 1$. The solution gives $x = 3$.

• Example 16.2: Solve $\sqrt{x - 1} - \sqrt{x - 6} = 1$. The solution gives $x = 10$.

Exercises 16.1 through 16.4: Solve each radical equation.

Lesson 17: Complex Numbers

Introduction to Complex Numbers

• Definition: The imaginary unit $i$ is defined as $i = \sqrt{-1}$, implying that $i^2 = -1$.

• Notation: A complex number is expressed as $a + bi$ where $a$ is the real part and $b$ is the imaginary part.

Operations on Complex Numbers

• Similar rules apply for addition, subtraction, and multiplication as with real numbers.

Examples

1. Example 17.1: Simplify expressions using the imaginary unit and perform operations on complex numbers.

   • Each expression must be simplified to the form $a + bi$.

Exercises 17.1 through 17.6: Various exercises on operations with complex numbers.

Lesson 18: Complete the Square

Completing the Square in Quadratics

• The square root property states if $X^2 = d$ then $X = \pm \sqrt{d}$.

• The process of completing the square allows solving quadratic equations.

Examples

1. Example 18.1: Solve $x^2 + 2x - 1 = 0$ by isolating the constant and completing the square.

   • Result: Solutions are $x = -1 \pm \sqrt{2}$.

Exercises 18.1 through 18.4: Solve the quadratic equations by completing the square.

Lesson 19: Quadratic Formula

The Quadratic Formula

• The quadratic formula is expressed as: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• The discriminant, $b^2 - 4ac$, determines the nature of the roots.

Examples

1. Example 19.1: Solve using the quadratic formula to find the number and type of solutions for a quadratic equation.

2. Example 19.2: Calculate the roots of the equation $2x^2 - 4x + 7 = 0$ using the above formula.

Exercises 19.1 through 19.6: Solve quadratic equations and real-life applications involving quadratics.

Radical Expressions

• Definition: If $b^2 = a$, then $b$ is called a square root of $a$.

• Notation: The positive square root of $a$ is denoted by $\sqrt{a}$, referred to as the principal square root.

• Function Definition: The function $f(x) = \sqrt{x}$ is termed a square root function.

• Domain: As a real-valued function, the domain consists of all real numbers $x$ such that $x \ge 0$, which can be expressed in interval notation as $[0, +\infty)$.

• Simplification: For any real number $a$, the expression $\sqrt{a^2}$ can be simplified to $\sqrt{a^2} = |a|$.

Cube Roots

• Definition: If $b^3 = a$, then $b$ is classified as a cube root of $a$.

• Notation: The cube root of a real number $a$ is denoted as $\sqrt[3]{a}$ .

• Simplification: For any real number $a$, the expression $\sqrt[3]{a^3}$ simplifies to $\sqrt[3]{a^3} = a$.

N-th Roots

• General Definition: If $b^n = a$, then $b$ is termed an n-th root of $a$.

• Notation: If $n$ is even, the positive n-th root is referred to as the principal n-th root, denoted by $\sqrt[n]{a}$ .

• If $n$ is odd, then the n-th root $\sqrt[n]{a}$ maintains the same sign as $a$ .

• Components:

  • Radical Sign: In $\sqrt[n]{a}$, the symbol $\sqrt{}$ is the radical sign.

  • Radicand: In $\sqrt[n]{a}$, $a$ is the radicand.

  • Index: In $\sqrt[n]{a}$, $n$ is referred to as the index.

• Restrictions: If $n$ is even, the n-th root of a negative number is not a real number.

• Simplification Rules:

  1. $\sqrt[n]{a^n} = |a|$ if $n$ is even.

  2. $\sqrt[n]{a^n} = a$ if $n$ is odd.

• Radical simplification means that the radicand has no perfect power factors that are applicable against the radical.

Examples of Radical Simplification

• Example 14.1: Simplify the radical expression using the definition.

  1. $\sqrt[4]{(y-1)^2}$

  • Solution: $\sqrt[4]{(y-1)^2} = ( (y-1)^2 )^{\frac{1}{4}} = (y-1)^{\frac{2}{4}} = (y-1)^{\frac{1}{2}} = \sqrt{y-1}$ (assuming $y-1 \ge 0$).

  1. $\sqrt[3]{-8x^3y^6}$

  • Solution:

    1. Identify the index: The index of the radical is $3$.

    2. Factor the radicand into perfect cubes: We need to rewrite each component of the radicand as a term raised to the power of $3$.

       • For $-8$: $-8 = (-2)^3$

       • For $x^3$: This is already a perfect cube.

       • For $y^6$: We can write $y^6 = (y^2)^3$ because $(a^m)^n = a^{mn}$.

    3. Rewrite the expression: Substitute the factored terms back into the radical:
       $\sqrt[3]{(-2)^3 \cdot x^3 \cdot (y^2)^3}$

    4. Apply the product rule for radicals: $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$.
       $\sqrt[3]{(-2)^3} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{(y^2)^3}$

    5. Simplify each term using the n-th root rule for odd n: Since the index $n=3$ is odd, $\sqrt[n]{a^n} = a$.

       • $\sqrt[3]{(-2)^3} = -2$

       • $\sqrt[3]{x^3} = x$

       • $\sqrt[3]{(y^2)^3} = y^2$

    6. Combine the simplified terms: Multiply the results from the previous step.
       $-2 \cdot x \cdot y^2 = -2xy^2$

Rational Exponents

• Definition: If $\sqrt[n]{a}$ is a real number, then we define a rational exponent as follows:

  • $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

Properties of Rational Exponents

1. $a^m \cdot a^n = a^{m+n}$.

2. $\frac{a^m}{a^n} = a^{m-n}$.

3. $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$ .

4. $(a^m)^n = a^{mn}$.

5. $(ab)^m = a^m \cdot b^m$.

6. $\sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}}$.

Examples of Simplifying Expressions with Rational Exponents

• Example 14.2:

  1. $\sqrt{x}\sqrt[3]{x^2}$

  • Solution: $\sqrt{x}\sqrt[3]{x^2} = x^{\frac{1}{2}} \cdot x^{\frac{2}{3}} = x^{\frac{1}{2}+\frac{2}{3}} = x^{\frac{7}{6}} = x\sqrt[6]{x}$.

  1. $\sqrt[3]{\sqrt{x^3}}$

  • Solution: $\sqrt[3]{(x^3)^{\frac{1}{2}}} = (x^{3})^{\frac{1}{6}} = x^{\frac{3}{6}} = x^{\frac{1}{2}} = \sqrt{x}$.

Exercises for Practice

Exercise 14.1: Evaluate the square root. If not a real number, state so.

1. $- \sqrt{\frac{4}{25}}$ (not a real number)

2. $\sqrt{49}-\sqrt{9}$

3. $- \sqrt{-1}$

Exercise 14.2: Simplify the radical expressions.

Exercise 14.3: Simplify the radical expressions.

Exercise 14.4: Simplify the radical expression, assuming all variables are positive.

Lesson 15: Algebra of Radicals

Product and Quotient Rules for Radicals

• If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, then the multiplication is as follows:

  $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$ .

• If $b \neq 0$, $\sqrt[n]{a} \div \sqrt[n]{b} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$.

Examples

1. Example 15.1: Simplify: $\sqrt[4]{8xy^4} \cdot \sqrt[4]{2x^7y}$ .

   • Solution: $\sqrt[4]{8xy^4 \cdot 2x^7y} = \sqrt[4]{16x^8y^5}$.

2. Example 15.2: Combine like radicals. $\sqrt{8x^3} - \sqrt{(-2)^2x^4} + \sqrt{2x^5}$:

   • Solution: $2x\sqrt{2x} - 2x^2 + x^2\sqrt{2x}$.

Rationalizing Denominators

• Definition: Rationalizing the denominator means rewriting a radical expression into an equivalent expression where the denominator has no radicals.

• Example 15.4:

  1. $\frac{1}{2\sqrt{x^3}}$

  • Multiply the expression by $\frac{\sqrt{x}}{\sqrt{x}}$.

  1. $\frac{\sqrt{x} \cdot 1}{2\sqrt{x^2}}$ .

Exercises

Exercise 15.1 through Exercise 15.9: Various exercises on simplifying, adding, subtracting, and rationalizing radical expressions.

Lesson 16: Solve Radical Equations

Techniques for Solving Radical Equations

• The process generally involves isolating the radical expression and taking the appropriate power of both sides to eliminate the radical.

Example Problems

• Example 16.1: Solve $x - \sqrt{x + 1} = 1$. The solution gives $x = 3$.

• Example 16.2: Solve $\sqrt{x - 1} - \sqrt{x - 6} = 1$. The solution gives $x = 10$.

Exercises

16.1 through 16.4: Solve each radical equation.

Lesson 17: Complex Numbers

Introduction to Complex Numbers

• Definition: The imaginary unit $i$ is defined as $i = \sqrt{-1}$, implying that $i^2 = -1$.

• Notation: A complex number is expressed as $a + bi$ where $a$ is the real part and $b$ is the imaginary part.

Operations on Complex Numbers

• Similar rules apply for addition, subtraction, and multiplication as with real numbers.

Examples

1. Example 17.1: Simplify expressions using the imaginary unit and perform operations on complex numbers.

   • Each expression must be simplified to the form $a + bi$ .

Exercises

17.1 through 17.6: Various exercises on operations with complex numbers.

Lesson 18: Complete the Square

Completing the Square in Quadratics

• The square root property states if $X^2 = d$ then $X = \pm \sqrt{d}$.

• The process of completing the square allows solving quadratic equations.

Examples

1. Example 18.1: Solve $x^2 + 2x - 1 = 0$ by isolating the constant and completing the square.

   • Result: Solutions are $x = -1 \pm \sqrt{2}$.

Exercises

18.1 through 18.4: Solve the quadratic equations by completing the square.

Lesson 19: Quadratic Formula

The Quadratic Formula

• The quadratic formula is expressed as: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• The discriminant, $b^2 - 4ac$, determines the nature of the roots.

Examples

1. Example 19.1: Solve using the quadratic formula to find the number and type of solutions for a quadratic equation.

2. Example 19.2: Calculate the roots