Precalculus Study Guide: Exponents, Radicals, Polynomials, and Rational Expressions

P2 Exponents

1. Product Rule

When multiplying expressions with the same base, add the exponents:

• Formula: $a^m \cdot a^n = a^{m+n}$

• Example: $x^3 \cdot x^2 = x^{3+2} = x^5$

2. Quotient Rule

When dividing expressions with the same base, subtract the exponents:

• Formula: $\frac{a^m}{a^n} = a^{m-n},\ a \neq 0$

• Example: $\frac{y^7}{y^3} = y^{7-3} = y^4$

3. Zero Exponent Rule

Any nonzero number raised to the power of zero equals 1:

• Formula: $a^0 = 1,\ a \neq 0$

• Example: $5^0 = 1$

4. Negative Exponent Rule

A negative exponent means the reciprocal:

• Formula: $a^{-n} = \frac{1}{a^n},\ a \neq 0$

• Example: $x^{-4} = \frac{1}{x^4}$

5. Power Rule

When raising a power to another power, multiply the exponents:

• Formula: $(a^m)^n = a^{mn}$

• Example: $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$

Radicals and Rational Exponents

1. Product Rule for Square Roots

The square root of a product is the product of the square roots:

• Formula: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b},\ a, b \geq 0$

• Example: $\sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}$

2. Quotient Rule for Square Roots

The square root of a quotient is the quotient of the square roots:

• Formula: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}},\ a \geq 0,\ b > 0$

• Example: $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$

3. Rationalizing Denominators

To simplify a radical expression, eliminate square roots in the denominator:

• Example: $\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$

• Example: $\frac{1}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b}$

P4-Polynomials

1. Identifying Polynomials, Standard Form, Degree, Leading Term, Leading Coefficient

A polynomial is an expression of the form $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where $a_i$ are real numbers and $n$ is a nonnegative integer.

• Standard Form: Terms are written in descending powers of $x$.

• Degree: The highest exponent of $x$.

• Leading Term: The term with the highest power of $x$.

• Leading Coefficient: The coefficient of the leading term.

• Example: For $f(x) = 7x^3 - 4x^2 + 2x - 5$, degree is 3, leading term is $7x^3$, leading coefficient is 7.

2. Multiplying Polynomials

Use the distributive property (each term in the first polynomial multiplies each term in the second).

• Special Cases:

  • $(a+b)^2 = a^2 + 2ab + b^2$

  • $(a-b)^2 = a^2 - 2ab + b^2$

  • $(a+b)(a-b) = a^2 - b^2$

• Example: $(x+2)(x+5) = x^2 + 7x + 10$

Factoring Polynomials

1. Factoring Out the Greatest Common Factor (GCF)

Find the largest factor common to all terms and factor it out.

• Example: $6x^2 + 9x^2 = 3x^2(2 + 3)$

2. Factoring by Grouping

Group terms in pairs, factor each group, then factor out the common binomial.

• Example: $x^3 + 3x^2 + 2x + 6 = (x^3 + 3x^2) + (2x + 6) = x^2(x + 3) + 2(x + 3) = (x^2 + 2)(x + 3)$

3. Factoring Trinomials

For $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$.

• Example: $x^2 + 7x + 6 = (x + 6)(x + 1)$

4. Difference of Squares Formula

Any expression of the form $a^2 - b^2$ can be factored as $(a - b)(a + b)$.

• Example: $x^2 - 25 = (x - 5)(x + 5)$

P6-Rational Expressions

Definition

A rational expression is a fraction where the numerator and denominator are polynomials:

• General Form: $\frac{P(x)}{Q(x)}$, $Q(x) \neq 0$

• Undefined: The expression is undefined wherever the denominator equals zero.

1. Multiplying Rational Expressions

Multiply the numerators together and denominators together, then simplify:

• Formula: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd},\ b, d \neq 0$

• Example: $\frac{x}{y} \cdot \frac{y}{z} = \frac{xy}{yz} = \frac{x}{z}$

2. Dividing Rational Expressions

Multiply by the reciprocal of the second fraction:

• Formula: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc},\ b, c, d \neq 0$

• Example: $\frac{x}{y} \div \frac{z}{w} = \frac{x}{y} \cdot \frac{w}{z} = \frac{xw}{yz}$

3. Adding or Subtracting Rational Expressions

Use the least common denominator (LCD):

• Formula: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

• Example: $\frac{1}{x} + \frac{2}{x+1} = \frac{(x+1) + 2x}{x(x+1)} = \frac{3x+1}{x(x+1)}$

4. Finding When Rational Expressions Are Undefined

A rational expression is undefined wherever its denominator equals zero.

• Example: $\frac{x+3}{x^2-4}$ is undefined at $x = \pm 2$

Summary Table: Exponent and Radical Rules

Rule	Formula	Example
Product Rule (Exponents)	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^3 = x^5$
Quotient Rule (Exponents)	$\frac{a^m}{a^n} = a^{m-n}$	$y^5 / y^2 = y^3$
Zero Exponent	$a^0 = 1$	$7^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = 1/x^2$
Power Rule	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Product Rule (Radicals)	$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$	$\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$
Quotient Rule (Radicals)	$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$	$\sqrt{\frac{9}{16}} = 3/4$

Key Takeaways

• Mastering exponent and radical rules is essential for simplifying algebraic expressions.

• Polynomials are classified by degree, leading term, and leading coefficient; always write in standard form.

• Factoring polynomials involves recognizing patterns such as GCF, grouping, trinomials, and difference of squares.

• Rational expressions require careful attention to domain restrictions (where denominators are zero).

• Operations with rational expressions mirror those with numerical fractions: find common denominators for addition/subtraction, multiply/divide as usual, and always simplify.