Evaluating a Negative Power

Evaluate $2^{-3}$.

Apply the negative exponent rule: $a^{-n} = \frac{1}{a^n}$ = $2^{-3} = \frac{1}{2^3}$

Calculate the power in the denominator: $2^3 = 2 \times 2 \times 2 = 8$ = $\frac{1}{8}$

Check the sign: The base $2$ is positive, so the result is positive = $\frac{1}{8} = 0.125$

Answer: $2^{-3} = \frac{1}{8}$, a small positive number, not $-8$.

Small Powers of 10

Write $10^{-3}$ as a fraction and as a decimal.

Apply the negative exponent rule: $10^{-3} = \frac{1}{10^3}$ = $\frac{1}{10^3}$

Calculate $10^3$: $10^3 = 1000$ = $\frac{1}{1000}$

Write as a decimal: One thousandth: the 1 sits in the third decimal place = $0.001$

Answer: $10^{-3} = \frac{1}{1000} = 0.001$. The exponent $-3$ tells you the 1 is in the third place after the decimal point.

A Fraction to a Negative Power

Evaluate $\left(\frac{2}{3}\right)^{-2}$.

Flip the fraction and make the exponent positive: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ = $\left(\frac{3}{2}\right)^{2}$

Square the numerator and the denominator: $\frac{3^2}{2^2} = \frac{9}{4}$ = $\frac{9}{4}$

Check with the definition: $\left(\frac{2}{3}\right)^{-2} = \frac{1}{\frac{4}{9}} = \frac{9}{4}$ = Same result

Answer: $\left(\frac{2}{3}\right)^{-2} = \frac{9}{4} = 2.25$

Moving Factors Across the Fraction Bar

Rewrite $\frac{x^{-3}}{y^{-2}}$ with positive exponents.

Move $x^{-3}$ from the numerator to the denominator: $x^{-3} = \frac{1}{x^3}$, so it becomes $x^3$ below the bar = $\frac{1}{x^3 \, y^{-2}}$

Move $y^{-2}$ from the denominator to the numerator: $\frac{1}{y^{-2}} = y^2$, so it becomes $y^2$ above the bar = $\frac{y^2}{x^3}$

Check that every exponent is positive: $y^2$ and $x^3$ = Done

Answer: $\frac{x^{-3}}{y^{-2}} = \frac{y^2}{x^3}$. A factor changes the sign of its exponent when it crosses the fraction bar.

Combining All the Exponent Rules

Simplify $\frac{6a^{-2}b^{3}}{3a^{4}b^{-1}}$ and write the answer with positive exponents.

Divide the coefficients: $6 \div 3 = 2$ = $2$

Quotient rule for $a$: $a^{-2-4} = a^{-6}$ = $a^{-6}$

Quotient rule for $b$ (subtracting a negative): $b^{3-(-1)} = b^{3+1} = b^{4}$ = $b^{4}$

Write with positive exponents: $2a^{-6}b^{4} = \frac{2b^{4}}{a^{6}}$ = $\frac{2b^{4}}{a^{6}}$

Answer: $\frac{6a^{-2}b^{3}}{3a^{4}b^{-1}} = \frac{2b^{4}}{a^{6}}$. Only $a^{-6}$ moves down; the coefficient $2$ stays on top.

Mistake: Thinking a negative exponent makes the number negative: $2^{-3} = -8$

Why: The minus sign is in the exponent, not in front of the number. It means "reciprocal", not "negative".

Correct: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$, a positive number.

Mistake: Moving the coefficient too: $2x^{-3} = \frac{1}{2x^3}$

Why: The exponent $-3$ belongs only to $x$. The $2$ has exponent $1$ and stays where it is.

Correct: $2x^{-3} = \frac{2}{x^3}$. Only write $\frac{1}{(2x)^3}$ when the problem says $(2x)^{-3}$ with parentheses.

Mistake: Flipping a fraction but forgetting the power, or making it negative: $\left(\frac{2}{3}\right)^{-2} = \frac{3}{2}$ or $-\frac{4}{9}$

Why: The minus sign flips the fraction, and the $2$ still squares it. Both parts of the exponent have a job.

Correct: $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}$

Mistake: Subtracting a negative exponent the wrong way: $\frac{x^5}{x^{-2}} = x^{3}$

Why: The quotient rule subtracts the whole exponent, including its sign: $5 - (-2)$, not $5 - 2$.

Correct: $\frac{x^5}{x^{-2}} = x^{5-(-2)} = x^{7}$

Tiny Measurements in Science

Scientists write very small lengths with negative powers of 10. The metric prefixes milli, micro and nano stand for $10^{-3}$, $10^{-6}$ and $10^{-9}$.

A millimeter is $10^{-3}$ m $= 0.001$ m, and a typical bacterium is about $2 \times 10^{-6}$ m long.

Camera Exposure Stops

Photographers measure light in stops. Each stop darker halves the light, so $n$ stops let in $2^{-n}$ of the original light.

A filter that is 3 stops dark lets in $2^{-3} = \frac{1}{8}$ of the light.

Diluting Samples in a Lab

In a lab, a sample is often diluted 1 to 10 several times. After $n$ such dilutions, the concentration is $10^{-n}$ times the original.

After 3 dilutions, a sample has $10^{-3} = 0.001$ of its original concentration.

Q: Why is $a^{-1}$ the reciprocal of $a$?

A: By the product rule, $a^{1} \times a^{-1} = a^{0} = 1$. The number you multiply by $a$ to get $1$ is $\frac{1}{a}$, so $a^{-1} = \frac{1}{a}$.

Q: Can a power with a negative exponent ever be negative?

A: Only if the base itself is negative. For example, $(-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}$. The negative exponent never adds a minus sign by itself.

Q: Why must the base not be zero?

A: $0^{-2}$ would mean $\frac{1}{0^2} = \frac{1}{0}$, and division by zero is not defined. That is why the rule says $a \neq 0$.

Q: Is it wrong to leave negative exponents in an answer?

A: $x^{-3}$ and $\frac{1}{x^3}$ are equal, so neither is wrong. Many teachers and textbooks ask for positive exponents in the final answer, so rewrite at the end unless the task says otherwise.