Exponents and Scientific Notation: Fundamental Concepts of Algebra (Precalculus Chapter P.2)

Prerequisites: Fundamental Concepts of Algebra

P.2 Exponents and Scientific Notation

This section introduces the foundational rules for working with exponents and scientific notation, which are essential for algebraic manipulation and problem-solving in precalculus and beyond.

Objectives

• Use properties of exponents.

• Simplify exponential expressions.

• Use scientific notation.

Properties of Exponents

Exponents are used to represent repeated multiplication of a base. The following rules are fundamental for simplifying and manipulating exponential expressions:

• Negative Exponent Rule: For any real number b (other than 0) and natural number n:

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

• Zero-Exponent Rule: For any real number b (other than 0):

$b^0 = 1$

• Product Rule: When multiplying exponential expressions with the same base, add the exponents:

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

• Power Rule (Powers to Powers): When an exponential expression is raised to a power, multiply the exponents:

$(b^m)^n = b^{mn}$

• Quotient Rule: When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator:

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

• Products-to-Powers Rule: When a product is raised to a power, raise each factor to that power:

$(ab)^n = a^n b^n$

• Quotients-to-Powers Rule: When a quotient is raised to a power, raise the numerator and denominator to that power:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$,   where $b \neq 0$

Simplifying Exponential Expressions

An exponential expression is considered simplified when:

• No parentheses appear.

• No powers are raised to powers.

• Each base occurs only once.

• No negative or zero exponents appear.

Example: Simplifying Exponential Expressions

• Example a: $2^4 \cdot x^3 \cdot 4 \cdot y^6 \cdot 16 \cdot x^{12} \cdot y^{24}$

• Example b: $\left(\frac{5x}{y^4}\right)^{-2} \cdot 5^{-2} x^{-2} 5^{-2} x^{-2} y^{-8} 5^{-2} x^{-2} y^8 / 25x^2$

Additional info: To simplify, apply the rules above, combine like terms, and ensure all exponents are positive and each base appears only once.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in the form:

$a \times 10^n$

where $1 \leq |a| < 10$ and $n$ is an integer.

Converting Between Decimal and Scientific Notation

• Decimal to Scientific Notation:

  • Move the decimal point to obtain a number between 1 and 10 (including 1).

  • The number of places moved determines the exponent $n$ on 10.

  • If the decimal is moved to the left, $n$ is positive; if to the right, $n$ is negative; if not moved, $n = 0$.

• Scientific Notation to Decimal:

  • Move the decimal point $n$ places to the right if $n$ is positive, or $|n|$ places to the left if $n$ is negative.

Examples

• Decimal to Scientific Notation:

  • $5,210,000,000 = 5.21 \times 10^9$

  • $-0.00000006893 = -6.893 \times 10^{-8}$

• Scientific Notation to Decimal:

  • $-2.6 \times 10^9 = -2,600,000,000$

  • $3.017 \times 10^{-6} = 0.000003017$

Computations with Scientific Notation

When multiplying or dividing numbers in scientific notation:

• Multiply or divide the numerical factors ($a$ values).

• Add exponents when multiplying powers of 10; subtract exponents when dividing.

Examples

• Multiplication: $(7.1 \times 10^5)(5 \times 10^{-7}) = (7.1 \times 5) \times 10^{5 + (-7)} = 35.5 \times 10^{-2} = 3.55 \times 10^{-1}$

• Division: $\frac{1.2 \times 10^6}{3.0 \times 10^{-3}} = \frac{1.2}{3.0} \times 10^{6 - (-3)} = 0.4 \times 10^{9} = 4 \times 10^8$