Rules of Exponents and Applications: College Algebra Study Notes

Rules of Exponents

Introduction to Exponents

An exponential expression is any expression containing one or more exponents. In the expression $a^n$, a is the base and n is the exponent or power. To evaluate an exponent, multiply the base by itself as many times as the value of the exponent. For example:

• $3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$

• $(-4)^2 = (-4) \times (-4) = 16$

Key Point: The difference between $-4^2$ and $(-4)^2$ is that $-4^2$ means $-(4^2) = -16$, while $(-4)^2 = 16$.

Rules of Exponents

Exponent rules allow us to simplify expressions involving powers. The main rules are summarized below:

Rule Name	Formula	Conditions
Product Rule	$a^m \cdot a^n = a^{m+n}$	$a \neq 0$
Quotient Rule	$\dfrac{a^m}{a^n} = a^{m-n}$	$a \neq 0$
Power Rule	$(a^m)^n = a^{mn}$	$a \neq 0$
Power of a Product	$(ab)^n = a^n b^n$	$a, b \neq 0$
Power of a Quotient	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	$a, b \neq 0$
Zero Exponent Rule	$a^0 = 1$	$a \neq 0$
Negative Exponent Rule	$a^{-n} = \dfrac{1}{a^n}$	$a \neq 0$

Product Rule

The Product Rule states that when multiplying like bases, add the exponents:

• Formula: $a^m \cdot a^n = a^{m+n}$

• Example: $x^2 \cdot x^5 = x^{2+5} = x^7$

Quotient Rule

The Quotient Rule states that when dividing like bases, subtract the exponents:

• Formula: $\dfrac{a^m}{a^n} = a^{m-n}$

• Example: $\dfrac{x^7}{x^3} = x^{7-3} = x^4$

Power Rule

The Power Rule states that when raising a power to another power, multiply the exponents:

• Formula: $(a^m)^n = a^{mn}$

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

Power of a Product

The Power of a Product Rule states that a product raised to a power equals each factor raised to that power:

• Formula: $(ab)^n = a^n b^n$

• Example: $(2x)^3 = 2^3 x^3 = 8x^3$

Power of a Quotient

The Power of a Quotient Rule states that a quotient raised to a power equals the numerator and denominator each raised to that power:

• Formula: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

• Example: $\left(\dfrac{x}{y}\right)^2 = \dfrac{x^2}{y^2}$

Zero Exponent Rule

The Zero Exponent Rule states that any nonzero base raised to the zero power is 1:

• Formula: $a^0 = 1$ for $a \neq 0$

• Example: $17^0 = 1$

Negative Exponent Rule

The Negative Exponent Rule states that a base raised to a negative exponent equals the reciprocal of the base raised to the positive exponent:

• Formula: $a^{-n} = \dfrac{1}{a^n}$

• Example: $4^{-3} = \dfrac{1}{4^3} = \dfrac{1}{64}$

Summary Table: Rules of Exponents

Rule	Formula
Product Rule	$a^m \cdot a^n = a^{m+n}$
Quotient Rule	$\dfrac{a^m}{a^n} = a^{m-n}$
Power Rule	$(a^m)^n = a^{mn}$
Power of a Product	$(ab)^n = a^n b^n$
Power of a Quotient	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
Zero Exponent Rule	$a^0 = 1$
Negative Exponent Rule	$a^{-n} = \dfrac{1}{a^n}$

Applications of Exponents

Scientific Notation

Scientific notation is a way to express very large or very small numbers using powers of ten. A number written in the form $a \times 10^n$ where $a$ is an integer greater than 0 and less than 10, and $n$ is any integer.

• To convert a number written in standard form to scientific notation:

  • Move the decimal point until the number is between 1 and 10.

  • The number of places moved is the exponent $n$.

  • If moved right, $n$ is negative. If moved left, $n$ is positive.

• To convert from scientific notation to standard form:

  • If $n$ is positive, move the decimal point to the right $n$ places.

  • If $n$ is negative, move the decimal point to the left $|n|$ places.

Example: $4.36 \times 10^7$ in standard form is $43,600,000$.

Operations with Scientific Notation

• Multiplication: Multiply the coefficients and add the exponents.

  • $(1.2 \times 10^4) \times (1.3 \times 10^5) = (1.2 \times 1.3) \times 10^{4+5} = 1.56 \times 10^9$

• Division: Divide the coefficients and subtract the exponents.

  • $\dfrac{5.4 \times 10^7}{3.2 \times 10^5} = \dfrac{5.4}{3.2} \times 10^{7-5} = 1.6875 \times 10^2$

Applied Problems

• Distance Example: The moon is approximately $2.4 \times 10^5$ miles from Earth. In standard form, this is $240,000$ miles.

• Economics Example: The GDP of the United States was $12,400,000,000,000$ in 2005. In scientific notation, this is $1.24 \times 10^{13}$.

• Social Media Example: If Facebook users upload 250 million photos daily, the total in a year (365 days) is $250,000,000 \times 365 = 9.125 \times 10^{10}$ photos.

Compound Interest

Compound Interest Formula

The compound interest formula calculates the amount of money in an account after a certain time period:

• Formula: $A = P \left(1 + \dfrac{r}{n}\right)^{nt}$

• Where:

  • $A$ = amount after $t$ years

  • $P$ = principal (initial amount)

  • $r$ = annual interest rate (as a decimal)

  • $n$ = number of times compounded per year

  • $t$ = number of years

• Example: If $P = 100$, $r = 0.07$, $n = 1$, $t = 4$, then $A = 100(1 + 0.07)^4$

Geometry and Exponents

Volume and Surface Area Formulas

• Rectangular Solid: $V = lwh$ where $l$ = length, $w$ = width, $h$ = height.

• Cylinder: $V = \pi r^2 h$ where $r$ = radius, $h$ = height.

• Example: For a cylinder with $r = 2$ ft and $h = 3$ ft, $V = \pi (2)^2 (3) = 12\pi$ ft$^3$

Concept Check

• If an exponent is not written, it is assumed to be 1 (e.g., $x = x^1$).

• In scientific notation, the standard form is $a \times 10^n$ where $a$ is between 1 and 10, not including 10.

• $4^2 = 16$

Practice Problems

• Simplify using exponent rules and write answers with positive exponents.

• Examples include:

  • $x^2 y^3 \cdot x^5 y^4$

  • $\dfrac{x^7}{x^3}$

  • $(2x^3 y^2)^4$

  • $\dfrac{a^7 b^2}{a^3 b^5}$

  • $\left(\dfrac{4a^2 b^3}{16a^4 b^7}\right)^{-2}$

Additional info: These notes cover College Algebra topics from Chapter 5 (Exponents and Polynomials) and include applications relevant to Chapters 10 and 12 (Rational Exponents, Radicals, and Exponential Functions).