6.7 Integer Exponents and Scientific Notation

Integer exponents and scientific notation give you ways to express and work with very large or very small numbers efficiently. These skills come up constantly in science and in later math courses, so getting comfortable with the rules now pays off.

Integer Exponents

Definition of negative exponents

A negative exponent tells you to take the reciprocal of the base raised to the positive version of that exponent. The general rule is:

$a^{-n} = \frac{1}{a^n}$, where $a \neq 0$

The base cannot be zero because you'd end up dividing by zero, which is undefined.

Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Think of it this way: a negative exponent doesn't make the answer negative. It flips the base to the bottom of a fraction.

Simplification with integer exponents

There are several exponent rules you need to know. Each one has a specific pattern for how the exponents behave.

Product Rule (same base, multiplying): Add the exponents. $a^m \cdot a^n = a^{m+n}$ Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

Quotient Rule (same base, dividing): Subtract the exponents. $\frac{a^m}{a^n} = a^{m-n}$ Example: $\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$

Power Rule (raising a power to a power): Multiply the exponents. $(a^m)^n = a^{mn}$ Example: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$

Product to a Power: Distribute the exponent to each factor. $(ab)^n = a^n b^n$ Example: $(2x)^3 = 2^3 x^3 = 8x^3$

Quotient to a Power: Distribute the exponent to the numerator and denominator. $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, where $b \neq 0$ Example: $\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$

Zero Exponent Rule: Any non-zero base raised to the power of 0 equals 1. $a^0 = 1$, where $a \neq 0$ Example: $5^0 = 1$

A common mistake is applying these rules when the bases are different. You can only add or subtract exponents when the bases match. $2^3 \cdot 3^4$ cannot be simplified using the product rule because the bases (2 and 3) aren't the same.

When simplifying expressions with multiple operations, follow the standard order of operations (PEMDAS): handle exponents before multiplication and division.

Scientific Notation

Decimal to scientific notation conversion

Scientific notation expresses a number as a product of two parts: a coefficient 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.

To convert a decimal number to scientific notation:

1. Move the decimal point until exactly one non-zero digit sits to its left.
2. Count how many places you moved the decimal. That count becomes your exponent $n$.
3. If you moved the decimal to the left, $n$ is positive (the original number was large). If you moved it to the right, $n$ is negative (the original number was small).

Examples:

• $3{,}670{,}000$: Move the decimal 6 places to the left → $3.67 \times 10^6$
• $0.00521$: Move the decimal 3 places to the right → $5.21 \times 10^{-3}$

A quick check: if the original number is bigger than 10, the exponent should be positive. If it's between 0 and 1, the exponent should be negative.

Operations in scientific notation

Multiplication: Multiply the coefficients and add the exponents. $(a \times 10^m) \cdot (b \times 10^n) = (a \cdot b) \times 10^{m+n}$ Example: $(2 \times 10^3) \cdot (3 \times 10^4) = 6 \times 10^7$

Division: Divide the coefficients and subtract the exponents. $\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$ Example: $\frac{6 \times 10^5}{2 \times 10^2} = 3 \times 10^3$

One thing to watch for: after multiplying or dividing, your coefficient might fall outside the 1-to-10 range. If that happens, you need to adjust. For instance, if you get $49.2 \times 10^{15}$, rewrite it as $4.92 \times 10^{16}$ by moving the decimal one place left and increasing the exponent by 1.

Scientific notation for real-world problems

When solving word problems with scientific notation, follow these steps:

1. Identify the given information and what you need to find.
2. Convert any numbers to scientific notation if they aren't already.
3. Perform the required operations using the multiplication/division rules above.
4. Adjust the coefficient so it falls between 1 and 10.
5. Express your final answer in whichever form the problem asks for (scientific notation or standard decimal).

Example: A light-year is approximately $9.46 \times 10^{15}$ meters. How far is 5.2 light-years in meters?

• Multiply: $(9.46 \times 10^{15}) \cdot 5.2 = 49.192 \times 10^{15}$
• The coefficient 49.192 is greater than 10, so adjust: $4.9192 \times 10^{16}$ meters.