Powers of Ten, Exponents, and Scientific Notation: A Concise Guide

Powers of Ten

A power of ten is a number obtained by raising 10 to an integer power.

General Form: $10^n$, where $n$ is an integer.

Positive Integer Exponents:

• $10^1 = 10$
• $10^2 = 10 \times 10 = 100$ (2 zeros)
• $10^3 = 10 \times 10 \times 10 = 1000$ (3 zeros)
• In general, $10^n$ (where $n > 0$) is 1 followed by $n$ zeros.

Zero Exponent:

• $10^0 = 1$ (Any non-zero number raised to the power of 0 is 1).

Negative Integer Exponents:

• $10^{-1} = \frac{1}{10^1} = 0.1$ (1 decimal place)
• $10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01$ (2 decimal places)
• $10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001$ (3 decimal places)
• In general, $10^{-n}$ (where $n > 0$) is 0 followed by $n-1$ zeros and then a 1.

Exponents

An exponent (or power) indicates how many times a base number is multiplied by itself.

General Form: $b^n$, where $b$ is the base and $n$ is the exponent.

Key Rules of Exponents:

• Product Rule: $b^m \times b^n = b^{m+n}$ (When multiplying with the same base, add the exponents).
• Quotient Rule: $\frac{b^m}{b^n} = b^{m-n}$ (When dividing with the same base, subtract the exponents).
• Power of a Power Rule: $(b^m)^n = b^{m \times n}$ (When raising a power to another power, multiply the exponents).
• Power of a Product Rule: $(ab)^n = a^n b^n$ (The power of a product is the product of the powers).
• Power of a Quotient Rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$ (The power of a quotient is the quotient of the powers).
• Negative Exponent Rule: $b^{-n} = \frac{1}{b^n}$ (A negative exponent indicates a reciprocal).
• Zero Exponent Rule: $b^0 = 1$ (Any non-zero base raised to the power of 0 is 1).

Metric Prefixes

Metric prefixes are used to scale the base units of the metric system by powers of ten. They provide a convenient way to express very large or very small quantities.

Common Metric Prefixes:

Prefix	Symbol	Power of Ten	Example
giga	G	$10^9$	1 gigameter (Gm) = $10^9$ m
mega	M	$10^6$	1 megawatt (MW) = $10^6$ W
kilo	k	$10^3$	1 kilometer (km) = $10^3$ m
centi	c	$10^{-2}$	1 centimeter (cm) = $10^{-2}$ m
milli	m	$10^{-3}$	1 millimeter (mm) = $10^{-3}$ m
micro	$\mu$	$10^{-6}$	1 micrometer ($\mu$m) = $10^{-6}$ m
nano	n	$10^{-9}$	1 nanosecond (ns) = $10^{-9}$ s
pico	p	$10^{-12}$	1 picofarad (pF) = $10^{-12}$ F

Metric prefixes directly utilize powers of ten, making conversions between different units within the metric system straightforward. For example, 5 kilometers is equal to $5 \times 10^3$ meters, or 5000 meters.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a concise and convenient form.

General Form: $a \times 10^n$, where:

• $1 \leq

  a

  < 10$ (The coefficient $a$ is a number between 1 and 10, including 1 but excluding 10).

• $n$ is an integer exponent (the power of ten).

Converting to Scientific Notation:

1. Move the decimal point in the original number until there is only one non-zero digit to the left of the decimal point.
2. Count the number of places the decimal point was moved. This number will be the absolute value of the exponent $n$.
3. If the decimal point was moved to the left, the exponent $n$ is positive.
4. If the decimal point was moved to the right, the exponent $n$ is negative.

Examples of Scientific Notation:

• $3,000,000 = 3 \times 10^6$
• $0.000052 = 5.2 \times 10^{-5}$
• $1.23 \times 10^4 = 12,300$
• $9.87 \times 10^{-2} = 0.0987$

Operations with Scientific Notation:

• Multiplication: $(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$
• Division: $\frac{a \times 10^m}{b \times 10^n} = (\frac{a}{b}) \times 10^{m-n}$
• Addition/Subtraction: Ensure the powers of ten are the same, then add or subtract the coefficients: $(a \times 10^n) \pm (b \times 10^n) = (a \pm b) \times 10^n$. If the powers of ten are different, adjust one of the numbers so that the powers are the same.