Powers with Integer Exponents
A video goes viral and its view count doubles every hour. After $h$ hours, the multiplier is $2^h$. That is easy to read forward in time. But what multiplier describes the moment the video was posted, when $h = 0$? And what about 3 hours before now, when $h = -3$? Until now, an exponent counted how many times you multiply. To handle $0$ and negative exponents, the meaning has to change: instead of multiplying, a negative exponent will divide.
Zero and Negative Exponents
Look at the powers of $2$ in order. Each time the exponent goes down by $1$, the value is divided by $2$. Keep the pattern going and see where it leads.
The same conclusion comes from the quotient rule. Dividing powers with the same base means subtracting exponents, and that rule has to keep working when the exponents are $0$ or negative. For example, $a^n \div a^n = a^{n-n} = a^0$. But any nonzero number divided by itself is $1$, so $a^0 = 1$. In the same way, $a^1 \div a^3 = a^{1-3} = a^{-2}$, while $\dfrac{a}{a^3} = \dfrac{1}{a^2}$.
Use the slider to move the exponent below zero and watch the value shrink toward $0$ without ever becoming negative.
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 $(-1)^{-4}$.
The Sign of a Power with a Negative Base
When the base is negative, the result can be positive or negative. A negative exponent first takes the reciprocal, so the sign comes from the base and the parity of the exponent (even or odd):
$$(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9} \qquad\qquad (-3)^{-3} = \frac{1}{(-3)^3} = -\frac{1}{27}$$Here the base can itself be written as $-a$, so the sign of the base depends on whether $a$ is positive or negative. Work out the sign of the base $-a$ first, then use the parity of the exponent.
Now organize all the cases (is $a$ positive or negative? is $n$ even or odd?) in a table with an example for each.
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.
Fractional Bases: Flip the Fraction
When the base is a fraction, a negative exponent flips the numerator and the denominator and makes the exponent positive.
The same reasoning with $a = 1$ tells you what $\left(\frac{1}{a}\right)^{-n}$ equals. Always give fraction answers in simplest form (lowest terms); if a problem says "irreducible fraction", it means exactly that.
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:
$\left[ \left( \frac{2}{3} \right)^{-2} \cdot \left( \frac{9}{4} \right)^{-1} \right]^{-3}$
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.
The Properties of Powers with Negative Exponents
The properties you already know still work when exponents are negative. Just add and subtract the exponents carefully, with their 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: $\left((-2)^3\right)^{-4}$.
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.
Writing Bases as Powers of Primes
Many expressions mix different bases, such as $8$ and $4$ or $12$ and $18$. To use the properties, rewrite every base as a product of prime powers first.
Simplify the following fraction by expressing both the numerator and the denominator as powers of the same prime base, and give the result as a single power:…
Simplify the following fraction by expressing both the numerator and the denominator as powers of the same prime base, and give the result as a single power: $\frac{8^4}{4^5}$.
Perform the necessary operations to express the following product as a single power with base $3$: $9^2 \cdot 27^{-1} \cdot 3^4$.
Perform the necessary operations to express the following product as a single power with base $3$: $9^2 \cdot 27^{-1} \cdot 3^4$.
Simplify the following expression, factoring the composite bases first and applying the properties of powers with integer exponents. Leave the result as a product of…
Simplify the following expression, factoring the composite bases first and applying the properties of powers with integer exponents. Leave the result as a product of powers of prime numbers:
$\frac{12^3 \cdot 18^{-2}}{6^4 \cdot 2^{-5}}$
Simplifying Expressions with Variables
With variables, every property works the same, and the goal is usually "positive exponents only." A factor with a negative exponent can move across the fraction bar and its exponent changes sign.
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:
$\frac{(a^{-2} \cdot b^3)^{-3} \cdot (a \cdot b^{-1} \cdot c^2)^4}{(a^3 \cdot c^{-5})^{-2}}$
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 $n$:
$\frac{4^{n+1} \cdot 10^{m-n}}{2^{m+n+2} \cdot 5^{m-n}}$
Exponential Equations
When the unknown is in the exponent, try to write both sides as powers of the same base. If $b^{u} = b^{v}$ (with $b > 0$, $b \ne 1$), then $u = v$.
When powers are added or subtracted, you cannot combine them directly. Instead, rewrite each with the common factor $b^x$ and factor it out.
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 factor 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 factor out a common factor:
$3^{x+1} + 3^{x-1} - 3^x = 21$
Modeling Growth and Decay
Many real situations repeat the same multiplication each step: a population doubles, a filter removes two thirds, a fractal keeps three of every four pieces. A repeated factor is a power.
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.
A company invests a principal of $\$5000$ in an investment fund that offers an annual compound interest rate of $4\%…
A company invests a principal of $\$5000$ in an investment fund that offers an annual compound interest rate of $4\%$. The correct mathematical formula to calculate the final balance after $t$ years is $C = 5000 \cdot (1,04)^t$. If due to a computer error the system calculates the balance 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 balance at $5$ years and the erroneous balance 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 panes of glass, 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 $3$ panes of glass, 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: For the viral video, the view multiplier is $2^h$. At posting time ($h = 0$) it is $2^0 = 1$, so you start with $1$ times the original count. Three hours before now it is $2^{-3} = \dfrac{1}{8}$ of the current count.