Powers with Integer Exponents
A video goes viral: it has 800 views at 3:00 p.m., and the number of views doubles every hour. At 4:00 p.m. it has $800 \cdot 2^1$ views, at 5:00 p.m. it has $800 \cdot 2^2$. But how many views did it have at 2:00 p.m., one hour before 3:00? Until now, an exponent counted how many times to multiply. Exponents that are $0$ or negative tell us what happens at "now" and in the past. They don't multiply, they divide.
Zero and negative exponents
Start with a pattern. Each step down the list divides by $2$, and the exponent drops by $1$. Keep the pattern going past $2^1$.
The pattern forces $2^0 = 1$ (that's $2 \div 2$) and then $2^{-1} = \frac{1}{2}$, $2^{-2} = \frac{1}{4}$, and so on. The same thing follows from the quotient rule. We know $a^n \div a^n = 1$, and the rule "subtract the exponents" gives $a^n \div a^n = a^{n-n} = a^0$. So $a^0$ has to be $1$.
Why does the definition make sense? Multiplying by $a^{-n}$ means dividing by $a^n$, which is exactly what the pattern above does one step at a time. All the exponent rules you already know keep working.
| Base | Exponent even | Exponent odd |
|---|---|---|
| positive, such as $3$ | $3^{-2} = \frac{1}{9} > 0$ | $3^{-3} = \frac{1}{27} > 0$ |
| negative, such as $-3$ | $(-3)^{-2} = \frac{1}{9} > 0$ | $(-3)^{-3} = -\frac{1}{27} < 0$ |
Calculate the numerical value of the following powers with negative exponents, expressing the result as an irreducible fraction or an integer: $3^{-2}$, $(-2)^{-3}$,…
Calculate the numerical value of the following powers with negative exponents, expressing the result as an irreducible fraction or an integer: $3^{-2}$, $(-2)^{-3}$, $5^{-1}$ and $(-1)^{-4}$.
Analyze and discuss with reasoning the resulting sign of the mathematical expression $(-a)^{-n}$ depending on two factors: whether the real number $a$ is strictly…
Analyze and discuss with reasoning the resulting sign of the mathematical expression $(-a)^{-n}$ depending on two factors: whether the real number $a$ is strictly positive or negative, and whether the integer $n$ is even or odd. Create a diagram or table with the four possible cases resulting from this combination and accompany it with a specific numerical example for each of the cases.
Fraction bases: flip the fraction
When the base is already a fraction, a negative exponent flips the numerator and the denominator.
Express the following powers with fractional bases with a positive exponent and then calculate their exact value: $\left(\frac{2}{5}\right)^{-3}$ and…
Express the following powers with fractional bases with a positive exponent and then calculate their exact value: $\left(\frac{2}{5}\right)^{-3}$ and $\left(-\frac{1}{4}\right)^{-2}$.
Calculate and simplify the following expression fully, expressing the final result as an irreducible fraction:…
Calculate and simplify the following expression fully, expressing the final result as an irreducible fraction:
$\left( \frac{3}{2} \right)^{-2} \cdot \left( \frac{4}{9} \right)^{-1} : \left( \frac{1}{2} \right)^{-3}$
Simplify the following expression with fractions and negative exponents, applying the properties step by step and giving the final result as an irreducible fraction…
Simplify the following expression with fractions and negative exponents, applying the properties step by step and giving the final result as an irreducible fraction without negative exponents:
$\left[ \left( \frac{2}{3} \right)^{-2} \cdot \left( \frac{9}{4} \right)^{-1} \right]^{-3}$
Same base: add, subtract, multiply exponents
The product, quotient and power-of-a-power rules work with negative exponents just as they do with positive ones. You only have to be careful with the signs.
Apply the properties of multiplication of powers with the same base to express the result as a single power: $7^4 \cdot 7^{-6} \cdot 7^5$.
Apply the properties of multiplication of powers with the same base to express the result as a single power: $7^4 \cdot 7^{-6} \cdot 7^5$.
Simplify the following expression by solving the operations in the numerator and the denominator, and express the final result as a single power:…
Simplify the following expression by solving the operations in the numerator and the denominator, and express the final result as a single power: $\frac{5^{-3} \cdot 5^6}{5^2}$.
Reduce the following expression to a single power by applying the power of a power property: $\left((-2)^3\right)^{-4}$.
Reduce the following expression to a single power by applying the power of a power property: $\left((-2)^3\right)^{-4}$.
Different bases: same exponent or prime factors
Sometimes the bases differ. Look first at what the powers have in common.
When there are several composite bases, break each one into primes, then collect the exponent of each prime separately.
Use the property of the product of powers with the same exponent to express the following calculation as a single power: $4^{-3} \cdot 2^{-3} \cdot 5^{-3}$.
Use the property of the product of powers with the same exponent to express the following calculation as a single power: $4^{-3} \cdot 2^{-3} \cdot 5^{-3}$.
In a math exam, a student has solved an operation by writing that $2^{-3} \cdot 5^{-3} = 10^{-6}$. Explain in detail what the student's conceptual error was, state the…
In a math exam, a student has solved an operation by writing that $2^{-3} \cdot 5^{-3} = 10^{-6}$. Explain in detail what the student's conceptual error was, state the correct property that should have been used and calculate the correct result as a fraction.
Simplify the following expression, factorizing the composite bases first and applying the properties of powers with integer exponents. Leave the result as a product of…
Simplify the following expression, factorizing the composite bases first and applying the properties of powers with integer exponents. Leave the result as a product of powers of prime numbers:
$\frac{12^3 \cdot 18^{-2}}{6^4 \cdot 2^{-5}}$
Expressions with variables
All the rules also work with letters. Remember to give the final answer with positive exponents only, unless told otherwise. To get rid of a negative exponent, move the factor across the fraction bar and change the sign of its exponent.
Sometimes the exponents themselves contain variables. The trick is to write every base in terms of primes, as before, and then compare exponents.
You may also be asked to prove an identity. A proof starts from the definition and uses only valid steps, ending with the other side of the equation.
Simplify the following algebraic expression fully, where $a$, $b$ and $c$ are non-zero real numbers. Apply the properties of powers and leave the final result expressed…
Simplify the following algebraic expression fully, where $a$, $b$ and $c$ are non-zero real numbers. Apply the properties of powers and leave the final result expressed exclusively with positive exponents:
$\frac{(a^{-2} \cdot b^3)^{-3} \cdot (a \cdot b^{-1} \cdot c^2)^4}{(a^3 \cdot c^{-5})^{-2}}$
Prove mathematically that, for any real number $a \neq 0$ and any integer $n$, the identity $\left(\frac{1}{a}\right)^{-n} = a^n$ holds. Use for your proof the strict…
Prove mathematically that, for any real number $a \neq 0$ and any integer $n$, the identity $\left(\frac{1}{a}\right)^{-n} = a^n$ holds. Use for your proof the strict definition of a power with a negative exponent and the properties of operations with fractions.
Simplify the following algebraic expression, in which $m$ and $n$ are arbitrary integers. After simplifying, check and justify whether the final result depends on the…
Simplify the following algebraic expression, in which $m$ and $n$ are arbitrary integers. After simplifying, check and justify whether the final result depends on the values taken by $m$ and $n$:
$\frac{4^{n+1} \cdot 10^{m-n}}{2^{m+n+2} \cdot 5^{m-n}}$
Growth and decay
Whenever a quantity is multiplied by the same factor each time, powers describe it. If you start with $A_0$ and multiply by $b$ each step, after $t$ steps you have $A_0 \cdot b^t$. A factor like $\frac{1}{3}$ can be written $3^{-1}$, so repeated division is a negative exponent.
A colony of bacteria doubles its population every hour. If at the initial moment of observation there are $150$ bacteria, write a mathematical expression using powers…
A colony of bacteria doubles its population every hour. If at the initial moment of observation there are $150$ bacteria, write a mathematical expression using powers that indicates the number of bacteria after $h$ hours have passed. How many bacteria will there be in the colony after $6$ hours?
In a chemical laboratory, a toxic substance contained in a liquid is being filtered. Each filter installed in the circuit reduces the amount of toxins to a third. If the…
In a chemical laboratory, a toxic substance contained in a liquid is being filtered. Each filter installed in the circuit reduces the amount of toxins to a third. If the initial amount of toxins is $81$ ounces, express with a power with a negative exponent the amount of toxin remaining after passing through $4$ successive filters and calculate its exact value.
A tennis tournament is played by single elimination (in each match, the loser is eliminated and the winner goes through to the next round). If $2^7$ players participate…
A tennis tournament is played by single elimination (in each match, the loser is eliminated and the winner goes through to the next round). If $2^7$ players participate in the first round, how many matches in total will have been played when the tournament ends and there is a single absolute winner? Express your reasoning using operations with powers.
Exponential equations
In an equation like $5^{x+2} \cdot 25^{-1} = 125^3$ the unknown is in the exponent. The key fact is that if $b^m = b^n$ (with $b > 0$, $b \ne 1$), then $m = n$. So the plan is to get the same base on both sides.
When the powers are added or subtracted, you can't just add the exponents. Instead, factor out the smallest common power.
Find the value of the integer $x$ that satisfies the following exponential equation, expressing all terms as powers of the same base: $5^{x+2} \cdot 25^{-1} = 125^3$
Find the value of the integer $x$ that satisfies the following exponential equation, expressing all terms as powers of the same base:
$5^{x+2} \cdot 25^{-1} = 125^3$
Solve the following exponential equation. To do this, apply the properties of powers with integer exponents to separate the bases and take out a common factor:…
Solve the following exponential equation. To do this, apply the properties of powers with integer exponents to separate the bases and take out a common factor:
$3^{x+1} + 3^{x-1} - 3^x = 21$
Powers in context: fractals, light and money
Here are three situations where a power with an integer exponent does the heavy lifting.
A shrinking fractal
Start with a filled triangle of area $1$. At each step, every filled triangle is split into four smaller triangles and the middle one is removed. Each step keeps $3$ of the $4$ pieces, so the area is multiplied by $\frac{3}{4}$.
The pattern is $1$, $\frac{3}{4}$, $\left(\frac{3}{4}\right)^2$, … so after $n$ steps the area is $A_0 \cdot r^n$ with $r = \frac{3}{4}$. Multiplying by a constant factor at each step always gives a power.
Light through tinted glass
If each pane lets through a fraction $r$ of the light, then $n$ panes let through $I_0 \cdot r^n$. The fraction that is lost is what's left over: $1 - r^n$.
Compound interest and negative exponents
A savings account with annual rate $r$ grows by a factor of $(1 + r)$ each year, so $P$ dollars become $P(1+r)^t$ after $t$ years. A negative exponent runs the clock backward.
A company invests a capital of $\$5000$ in an investment fund that offers an annual compound interest of $4\%…
A company invests a capital of $\$5000$ in an investment fund that offers an annual compound interest of $4\%$. The correct mathematical formula to calculate the final capital after $t$ years is $C = 5000 \cdot (1,04)^t$. If due to a computer error the system calculates the capital using a negative exponent, $C_{erroneo} = 5000 \cdot (1,04)^{-t}$, what financial meaning would this erroneous calculation have? Furthermore, deduce what exact mathematical relationship exists between the correct capital at $5$ years and the erroneous capital calculated for those same $5$ years.
A geometric fractal called the "Sierpinski Triangle" is constructed starting from a blue-filled equilateral triangle whose initial area is $1$ unit. In each iteration…
A geometric fractal called the "Sierpinski Triangle" is constructed starting from a blue-filled equilateral triangle whose initial area is $1$ unit. In each iteration (step), each blue triangle is divided into four smaller equilateral triangles and the central triangle is removed (leaving it empty/black). Observe the figure illustrating the first steps. Determine the mathematical expression, in the form of a power, that represents the remaining blue area after $n$ iterations. From that expression, what will be the exact blue area after $6$ iterations?
The intensity of a beam of light when passing through a certain type of tinted glass is reduced to $\frac{3}{4}$ of its incident value. If we superimpose $n$ of these…
The intensity of a beam of light when passing through a certain type of tinted glass is reduced to $\frac{3}{4}$ of its incident value. If we superimpose $n$ of these identical glasses, the intensity of the resulting light that manages to pass through them is $I = I_0 \cdot \left(\frac{3}{4}\right)^n$, where $I_0$ is the initial intensity. Express the fraction of light that is lost (not what manages to pass) after passing through $3$ glasses, using powers with integer exponents for your calculations. In view of the result, determine whether the system blocks more or less than half of the original light.
Back to the start: the video had $800$ views at 3:00 p.m. and its views double every hour. One hour earlier it had $800 \cdot 2^{-1} = 400$ views, two hours earlier $800 \cdot 2^{-2} = 200$, and three hours earlier $800 \cdot 2^{-3} = 100$. At 3:00 p.m. itself the exponent is $0$, and $800 \cdot 2^0 = 800$.