Exponents and Scientific Notation: College Algebra Study Notes

Exponents and Scientific Notation

Introduction

This section introduces the fundamental rules and properties of exponents, including the product, quotient, zero, and negative exponent rules. It also covers scientific notation, a method for expressing very large or very small numbers efficiently. Mastery of these concepts is essential for success in College Algebra and related fields.

Exponents

Exponents are a shorthand notation for repeated multiplication of the same factor. They are used extensively in algebra to simplify expressions and solve equations.

• Definition: For any real number a and natural number n, an means multiplying a by itself n times.

• Base: The number being multiplied.

• Exponent (Power): Indicates how many times the base is used as a factor.

• Order of Operations: Exponents are evaluated before multiplication, division, addition, or subtraction.

Example: $3^4 = 3 \times 3 \times 3 \times 3 = 81$

The Product Rule for Exponents

The product rule allows you to simplify expressions where the same base is multiplied with different exponents.

• Rule: For any real number a and integers m and n:

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

• Application: Add the exponents when multiplying like bases.

Example:

• $3^2 \cdot 3^4 = 3^{2+4} = 3^6 = 729$

• $z^2 \cdot z^5 = z^{2+5} = z^{7}$

• $3y^2(-4y^4) = 3 \cdot y^2 \cdot (-4) \cdot y^4 = -12y^6$

Zero Exponent Rule

Any nonzero number raised to the zero power equals one.

• Rule: If a ≠ 0, then $a^0 = 1$

Example:

• $5^0 = 1$

• $(x^0)(y^0)(z^3)^0 = 1 \cdot 1 \cdot 1 = 1$

• $-(x^0) = -1$

The Quotient Rule for Exponents

The quotient rule is used when dividing like bases with different exponents.

• Rule: For any nonzero real number a and integers m and n:

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

• Application: Subtract the exponent in the denominator from the exponent in the numerator.

Example:

• $\frac{6^8}{6^2} = 6^{8-2} = 6^6$

• $\frac{z^7}{z^5} = z^{7-5} = z^2$

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent.

• Rule: For any nonzero real number a and positive integer n:

$a^{-n} = \frac{1}{a^n}$

• Application: Move the base with a negative exponent to the denominator and make the exponent positive.

Example:

• $x^{-3} = \frac{1}{x^3}$

• $(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$

Scientific Notation

Scientific notation is a method for writing very large or very small numbers as a product of a number between 1 and 10 and a power of 10.

• Form: $a \times 10^n$ where $1 \leq a < 10$ and $n$ is an integer.

• Purpose: Used in science and engineering to simplify calculations and representation of extreme values.

Steps to Write in Scientific Notation:

1. Move the decimal point in the original number until the new number is between 1 and 10.

2. Count the number of places the decimal was moved; this is the exponent on 10.

3. If the original number is greater than 10, the exponent is positive. If less than 1, the exponent is negative.

Examples:

• 4700: Move decimal 3 places left → $4.7 \times 10^3$

• 0.00047: Move decimal 4 places right → $4.7 \times 10^{-4}$

Converting Scientific Notation to Standard Form:

• Move the decimal point the number of places indicated by the exponent.

• Positive exponent: move right; negative exponent: move left.

Examples:

• $5.2738 \times 10^3 = 5273.8$

• $6.45 \times 10^{-5} = 0.0000645$