1.2 Exponents and Scientific Notation

Exponents give you a shorthand for repeated multiplication, and the rules governing them let you simplify expressions that would otherwise be tedious to work with. Scientific notation builds on exponents to express very large or very small numbers in a compact form you'll see constantly in science courses.

Exponent Rules and Properties

Exponent rules for operations

These three rules are the foundation. Every complex simplification you'll do comes back to applying one or more of them.

• Product rule: When you multiply two powers with the same base, add the exponents.

$x^a \cdot x^b = x^{a+b}$ Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

• Quotient rule: When you divide two powers with the same base, subtract the exponents.

$\frac{x^a}{x^b} = x^{a-b}$ Example: $\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$

• Power rule: When you raise a power to another power, multiply the exponents.

$(x^a)^b = x^{a \cdot b}$ Example: $(y^2)^3 = y^{2 \cdot 3} = y^6$

A common mistake is applying the product rule when the bases are different. $2^3 \cdot 3^4$ cannot be combined into a single base using the product rule because the bases (2 and 3) don't match.

Zero and negative exponents

• Zero exponent rule: Any non-zero base raised to the power 0 equals 1.

$x^0 = 1 \quad \text{for } x \neq 0$ This applies to any expression in the base: $4^0 = 1$, and $\left(\frac{2}{3}\right)^0 = 1$.

• Negative exponent rule: A negative exponent means "take the reciprocal," then apply the positive exponent.

$x^{-a} = \frac{1}{x^a} \quad \text{for } x \neq 0$ Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

• Combining negative exponents with other rules: You can use the negative exponent rule to rewrite terms, then simplify.

$\frac{3^{-2}}{4^{-1}} = \frac{\frac{1}{3^2}}{\frac{1}{4^1}} = \frac{1}{9} \cdot \frac{4}{1} = \frac{4}{9}$

Powers of products and quotients

• Power of a product: Distribute the exponent to every factor inside the parentheses.

$(xy)^n = x^n \cdot y^n$ Example: $(3a)^2 = 3^2 \cdot a^2 = 9a^2$

• Power of a quotient: Distribute the exponent to both the numerator and the denominator.

$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$ Example: $\left(\frac{2}{b}\right)^3 = \frac{2^3}{b^3} = \frac{8}{b^3}$

Complex exponential expressions

When you face an expression with multiple rules at play, work through it one step at a time. Here's a full walkthrough:

Simplify $(4x^2y^{-3})^3 \cdot (2x^{-1}y^2)^{-2}$

1. Apply the power rule to the first group: $(4x^2y^{-3})^3 = 4^3 \cdot x^{2 \cdot 3} \cdot y^{-3 \cdot 3} = 64x^6y^{-9}$

2. Apply the power rule to the second group. The outer exponent is $-2$, so multiply each inner exponent by $-2$: $(2x^{-1}y^2)^{-2} = 2^{-2} \cdot x^{(-1)(-2)} \cdot y^{(2)(-2)} = \frac{1}{4} \cdot x^2 \cdot y^{-4}$

3. Multiply the two results using the product rule (add exponents on matching bases): $64x^6y^{-9} \cdot \frac{1}{4}x^2y^{-4} = 16x^{6+2}y^{-9+(-4)} = 16x^8y^{-13}$

4. Rewrite with positive exponents (if required): $16x^8y^{-13} = \frac{16x^8}{y^{13}}$

Scientific Notation and Related Concepts

Standard vs. scientific notation

Scientific notation expresses a number as the product of a coefficient and a power of 10:

$a \times 10^n, \quad \text{where } 1 \leq |a| < 10 \text{ and } n \text{ is an integer}$

The coefficient $a$ must be at least 1 but less than 10. That's what distinguishes scientific notation from just "any expression with a power of 10."

To convert, move the decimal point until you have a number between 1 and 10, then count how many places you moved it:

• Large numbers: Move the decimal left. The exponent is positive.

$5{,}670{,}000 = 5.67 \times 10^6$ (decimal moved 6 places left)

• Small numbers: Move the decimal right. The exponent is negative.

$0.00092 = 9.2 \times 10^{-4}$ (decimal moved 4 places right)

Standard form is just the regular way of writing the number out with all its digits.

Calculations in scientific notation

Multiplication: Multiply the coefficients and add the exponents. $(a \times 10^n) \cdot (b \times 10^m) = (a \cdot b) \times 10^{n+m}$ Example: $(4 \times 10^2) \cdot (2 \times 10^4) = 8 \times 10^6$

If the product of the coefficients is 10 or greater, adjust. For instance, $(5 \times 10^3) \cdot (3 \times 10^2) = 15 \times 10^5 = 1.5 \times 10^6$.

Division: Divide the coefficients and subtract the exponents. $\frac{a \times 10^n}{b \times 10^m} = \left(\frac{a}{b}\right) \times 10^{n-m}$ Example: $\frac{8 \times 10^5}{2 \times 10^3} = 4 \times 10^2$

Interpreting scientific notation

To compare two numbers in scientific notation, look at the exponents first. A larger exponent means a larger magnitude, regardless of the coefficient. For example, $3.6 \times 10^9$ is greater than $7.2 \times 10^7$ because $10^9$ is 100 times larger than $10^7$.

If the exponents are the same, then compare the coefficients directly.

The exponent tells you roughly how many places the decimal moves, not exactly how many zeros appear at the end. For instance, the mass of Earth is approximately $5.97 \times 10^{24}$ kg. The exponent 24 means the decimal in 5.97 moves 24 places to the right, giving a number with 25 digits total.

When doing calculations, pay attention to significant figures. Your answer shouldn't have more precision than the least precise value you started with.

Related notations

• Engineering notation works like scientific notation but restricts the exponent to multiples of 3 (matching metric prefixes like kilo, mega, giga). For example, $47{,}000 = 47 \times 10^3$ in engineering notation.
• Logarithms are the inverse of exponentiation. If $10^n = x$, then $\log_{10}(x) = n$. You'll encounter these later in the course when solving exponential equations.

Applications of exponents

Exponents show up whenever a quantity grows or shrinks by repeated multiplication. Here's a classic example:

Problem: A bacteria population starts at 100 cells and doubles every 4 hours. How many bacteria are present after 24 hours?

1. Write the growth equation: $P = P_0 \cdot 2^{t/4}$, where $P_0 = 100$ and $t$ is time in hours.
2. Plug in $t = 24$: $P = 100 \cdot 2^{24/4} = 100 \cdot 2^6$
3. Evaluate: $2^6 = 64$, so $P = 100 \cdot 64 = 6{,}400$ bacteria.

In scientific notation, that's $6.4 \times 10^3$, which is convenient if the numbers get much larger (and in biology, they often do).