6.7 Integer Exponents and Scientific Notation

Integer exponents and scientific notation are powerful tools for expressing and manipulating numbers. They simplify calculations with very large or small values, making them essential in math and science.

These concepts build on basic arithmetic, expanding our ability to work with complex numbers. Understanding exponents and scientific notation opens doors to advanced math and real-world applications in fields like astronomy and physics.

Integer Exponents

Definition of negative exponents

• Negative exponents represent the reciprocal of the base raised to the positive exponent ($a^{-1} = \frac{1}{a}$)
• General form: $a^{-n} = \frac{1}{a^n}$, where $a \neq 0$ (0 cannot be raised to a negative exponent)
• Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Simplification with integer exponents

• Multiply powers with the same base by adding exponents: $a^m \cdot a^n = a^{m+n}$ ($2^3 \cdot 2^4 = 2^7$)
• Divide powers with the same base by subtracting exponents: $\frac{a^m}{a^n} = a^{m-n}$ ($\frac{3^5}{3^2} = 3^3$)
• Raise a power to another power by multiplying exponents: $(a^m)^n = a^{mn}$ ($(2^3)^4 = 2^{12}$)
• Raise each factor of a product to a power: $(ab)^n = a^n b^n$ ($(2x)^3 = 2^3x^3$)
• Raise numerator and denominator of a quotient to a power: $(\frac{a}{b})^n = \frac{a^n}{b^n}$, $b \neq 0$ ($(\frac{3}{4})^2 = \frac{3^2}{4^2}$)
• Any non-zero base raised to power of 0 equals 1: $a^0 = 1$, $a \neq 0$ ($5^0 = 1$)
• Follow the order of operations (PEMDAS) when simplifying expressions with exponents

Scientific Notation

Decimal to scientific notation conversion

• Express large or small numbers as a product of a number between 1 and 10 and a power of 10
• General form: $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer
• Steps to convert decimal to scientific notation:
  1. Move decimal point until one non-zero digit is left of it
  2. Count places decimal point moved (becomes exponent $n$)
  3. If moved left, $n$ is positive; if moved right, $n$ is negative
• Example: 0.00521 = $5.21 \times 10^{-3}$, 3,670,000 = $3.67 \times 10^6$
• Consider significant figures when converting between decimal notation and scientific notation

Operations in scientific notation

• Multiply numbers by multiplying coefficients and adding exponents: $(a \times 10^m) \cdot (b \times 10^n) = (ab) \times 10^{m+n}$
  • $(2 \times 10^3) \cdot (3 \times 10^4) = 6 \times 10^7$
• Divide numbers by dividing coefficients and subtracting exponents: $\frac{a \times 10^m}{b \times 10^n} = (\frac{a}{b}) \times 10^{m-n}$
  • $\frac{6 \times 10^5}{2 \times 10^2} = 3 \times 10^3$

Scientific notation for real-world problems

1. Identify given information and quantity to find
2. Convert numbers to scientific notation if needed
3. Perform required operations using scientific notation rules
4. Express final answer in scientific notation or decimal form based on context

• Example: A light-year is $9.46 \times 10^{15}$ meters. How many meters is 5.2 light-years?
  • Given: 1 light-year = $9.46 \times 10^{15}$ m, distance = 5.2 light-years
  • Multiply: $(9.46 \times 10^{15}) \cdot 5.2 = 49.192 \times 10^{15}$
  • Answer: 5.2 light-years is $4.9192 \times 10^{16}$ meters

Additional Concepts

Number Systems and Logarithms

• Different number systems (e.g., binary, decimal) use scientific notation for large or small values
• Logarithms are the inverse operations of exponents and are used to solve exponential equations
• The relationship between logarithms and scientific notation: $\log_{10}(a \times 10^n) = \log_{10}(a) + n$