10.5 Integer Exponents and Scientific Notation

Properties of Exponents and Scientific Notation

Definition of negative exponents

A negative exponent tells you to take the reciprocal of the base raised to the positive version of that exponent. For any nonzero number $a$ and integer $n$:

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

So $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$. The base doesn't become negative; it just moves to the denominator.

When negative and positive exponents appear in the same expression, only the factors with negative exponents move. For example:

$3x^{-2}y^3 = \frac{3y^3}{x^2}$

Here, $x^{-2}$ flips to the denominator as $x^2$, while $3$ and $y^3$ stay in the numerator because their exponents are already positive.

Also worth remembering: any nonzero number raised to the zero power equals 1. That is, $a^0 = 1$. This follows directly from the division rule (covered next), since $\frac{a^n}{a^n} = a^{n-n} = a^0 = 1$.

Simplification with integer exponents

Three core laws let you simplify expressions with exponents. All of them require the bases to match.

Product rule (same base, multiply): Add the exponents.

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

Quotient rule (same base, divide): Subtract the exponents.

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

Power rule (raising a power to a power): Multiply the exponents.

$(a^m)^n = a^{m \cdot n}$

To simplify a complex expression, combine like bases using these rules, then evaluate. A common mistake is trying to add exponents when the bases are different. $2^3 \cdot 3^4$ cannot be simplified with the product rule because the bases (2 and 3) don't match.

Scientific Notation and Its Applications

Decimal vs. scientific notation

Scientific notation is a shorthand for writing very large or very small numbers. The format is always:

$a \times 10^n$

where $a$ is a number with exactly one nonzero digit before the decimal point (so $1 \leq a < 10$) and $n$ is an integer.

For example, $420{,}000 = 4.2 \times 10^5$ and $0.00037 = 3.7 \times 10^{-4}$.

Converting decimal to scientific notation:

1. Move the decimal point until you have a number between 1 and 10.
2. Count how many places you moved it.
3. If you moved the decimal left, the exponent is positive. If you moved it right, the exponent is negative.

Converting scientific notation to decimal:

Move the decimal point the number of places indicated by the exponent. Move right for a positive exponent, left for a negative exponent, filling in zeros as needed.

Operations in scientific notation

Multiplying:

1. Multiply the coefficients (the numbers in front).
2. Add the exponents of 10.
3. Adjust the result so the coefficient is between 1 and 10.

If multiplying the coefficients gives something 10 or above, shift the decimal and increase the exponent by 1. For instance, $(5.0 \times 10^3)(4.0 \times 10^2) = 20.0 \times 10^5 = 2.0 \times 10^6$.

Dividing:

1. Divide the coefficients.
2. Subtract the exponent of the denominator from the exponent of the numerator.
3. Adjust if needed so the coefficient is between 1 and 10.

Applications of scientific notation

Scientific notation shows up whenever numbers are extremely large or extremely small. The distance from Earth to the Sun is about $1.5 \times 10^8$ kilometers. A hydrogen atom has a diameter of roughly $1.2 \times 10^{-10}$ meters. Writing these in standard form would mean tracking many zeros, which invites errors.

To solve a word problem with scientific notation, first convert any given values into scientific notation, then perform the required operations using the multiplication or division steps above. Convert your answer back to decimal form if the problem asks for it, and always include units.

Precision and Magnitude in Scientific Notation

The coefficient (sometimes called the mantissa) carries the significant digits of your number. It tells you how precise the measurement is. For example, $3.00 \times 10^5$ (three significant figures) is more precise than $3 \times 10^5$ (one significant figure), even though both represent 300,000.

The order of magnitude is determined by the exponent and tells you the general size of a number. Two numbers with the same exponent are in the same ballpark. Comparing orders of magnitude is a quick way to see how quantities stack up: something at $10^8$ is about a thousand times larger than something at $10^5$, since $10^{8-5} = 10^3 = 1000$.