Properties of Exponents
Two videos are posted at the same time, each with 1 view. Video A's views triple every hour. Video B's views multiply by 8 every two hours. After 10 hours, A has $3^{10}$ views and B has $8^5$. Which is bigger? Try to decide without a calculator.
You already know that $2^3 \cdot 2^5 = 2^8$. In Algebra 2 the hard part is not the rule itself. It is spotting which property to use inside a longer expression: bases that must be rewritten, variables, or an unknown hiding in the exponent. This page trains that skill. We'll come back to the videos at the end of the "Rewriting with a common base" section.
Product, quotient and power of a power
Start with what an exponent means: it counts repeated factors. Here is $2^3 \cdot 2^2$:
Put three factors of 2 next to two more factors of 2 and you have five factors of 2. That is the whole idea behind the first rule.
Express as a single power with base $5$ by applying the properties of powers: $\frac{5^4 \cdot 5^{-2}}{5^3}$.
Express as a single power with base $5$ by applying the properties of powers: $\frac{5^4 \cdot 5^{-2}}{5^3}$.
Reduce to a single power using the appropriate properties: $\left((-7)^3\right)^{-4} \cdot (-7)^5$.
Reduce to a single power using the appropriate properties: $\left((-7)^3\right)^{-4} \cdot (-7)^5$.
Zero and negative exponents
What should $2^0$ or $2^{-3}$ mean? Look for a pattern. Each step to the right divides by 2:
The pattern says $2^0 = 1$, $2^{-1} = \frac{1}{2}$, $2^{-2} = \frac{1}{4}$. The quotient rule agrees: $\dfrac{a^3}{a^3} = 1$ and also $a^{3-3} = a^0$, so $a^0$ must equal $1$.
Calculate the value of the following power and express the result as an irreducible fraction: $\left(\frac{2}{3}\right)^{-3}$.
Calculate the value of the following power and express the result as an irreducible fraction: $\left(\frac{2}{3}\right)^{-3}$.
Express the number $\frac{1}{128}$ as a power with base $2$ using a negative exponent.
Express the number $\frac{1}{128}$ as a power with base $2$ using a negative exponent.
Same exponent, different bases
Now the other situation: the bases are different but the exponent is the same.
This is also why scaling a shape works the way it does. Tripling the side of a square makes the area $(3s)^2 = 3^2 s^2 = 9s^2$, nine times bigger:
For a cube the exponent is 3. If every edge is multiplied by $k$, the volume is multiplied by $k^3$.
Write as a single power by applying the properties of operations with powers of the same exponent: $15^4 : 3^4 \cdot 2^4$.
Write as a single power by applying the properties of operations with powers of the same exponent: $15^4 : 3^4 \cdot 2^4$.
State whether the following statement is true or false, justifying your answer using the properties of powers: "Twice $2^8$ is $4^8$".
State whether the following statement is true or false, justifying your answer using the properties of powers: "Twice $2^8$ is $4^8$".
A packaging company manufactures cubic boxes. If they decide to triple the edge length of a standard box, by what factor will its volume be multiplied? Express your…
A packaging company manufactures cubic boxes. If they decide to triple the edge length of a standard box, by what factor will its volume be multiplied? Express your reasoning using the properties of the power of a product.
Simplifying expressions with variables
Variables work exactly like numbers. Assume every variable in a denominator is nonzero. A typical goal is a final answer with positive exponents only.
Exponents can contain variables too. Treat the whole exponent as one number, and use parentheses when you subtract it.
Simplify the following algebraic expression leaving the result with positive exponents: $\frac{x^3 \cdot y^{-2}}{x^{-1} \cdot y^4}$.
Simplify the following algebraic expression leaving the result with positive exponents: $\frac{x^3 \cdot y^{-2}}{x^{-1} \cdot y^4}$.
Simplify the following algebraic expression fully, using the properties of powers and leaving the result without negative exponents:…
Simplify the following algebraic expression fully, using the properties of powers and leaving the result without negative exponents: $\left( \frac{a^{-2} \cdot b^3}{a \cdot b^{-1}} \right)^{-2}$.
Simplify the following algebraic expression, where $x$ and $y$ are non-zero real numbers and $a, b$ are positive integers:…
Simplify the following algebraic expression, where $x$ and $y$ are non-zero real numbers and $a, b$ are positive integers:
$\left( \frac{x^{a-b} \cdot y^{b+a}}{x^{a+b} \cdot y^{b-a}} \right)^{-1}$
Apply the properties of powers with the same base and express the final result using only positive exponents.
Rewriting with a common base
When the bases are different numbers, rewrite them so they match. Break each base into prime factors, or recognize it as a power of a smaller number.
The same trick lets you compare big numbers without a calculator. Once the bases match, compare the exponents.
Now back to the two videos. We can make the exponents match instead of the bases.
Simplify the following expression by factoring the bases first and applying the properties of powers. Express the result as a product of powers with prime bases:…
Simplify the following expression by factoring the bases first and applying the properties of powers. Express the result as a product of powers with prime bases: $\frac{18^2 \cdot 12^{-3}}{24^{-1} \cdot 9^3}$.
Without using a calculator, determine which of the following quantities is greater: $3^{50}$ or $9^{20}$. Justify your answer by transforming one of the powers so that…
Without using a calculator, determine which of the following quantities is greater: $3^{50}$ or $9^{20}$. Justify your answer by transforming one of the powers so that both have the same base.
A student has shown the following algebraic working to simplify an expression:…
A student has shown the following algebraic working to simplify an expression:
$\frac{10^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = \frac{(2 \cdot 5)^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = \frac{2^4 \cdot 5^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = 5^{4 - 2 - 2} = 5^0 = 1$
Identify if there is any error in their reasoning, explain it in detail and provide the correct result by strictly applying the properties of powers.
Sums and differences of powers
There is no property for $a^m + a^n$. When exponents contain a variable, factor out the common power. This is just the distributive property.
Graph the expression and you see the same thing. It is a horizontal line, because the value does not depend on $x$:
Simplify the following expression in which the exponents depend on the parameter $n$, factoring out a common factor if necessary to be able to operate:…
Simplify the following expression in which the exponents depend on the parameter $n$, factoring out a common factor if necessary to be able to operate: $\frac{3^{n+2} - 3^n}{3^{n+1} + 3^n}$.
Exponential equations
An exponential equation has the unknown in the exponent. The key fact: if $b > 0$ and $b \neq 1$, then $b^m = b^n$ means $m = n$. So the plan is:
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Write both sides with the same base.
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Simplify each side using the properties.
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Set the exponents equal and solve the resulting equation. It is usually a linear equation.
Graphically, $2^x = \frac{1}{8}$ is where the curve $y = 2^x$ meets the line $y = \frac{1}{8}$:
They cross at $x = -3$, because $\frac{1}{8} = 2^{-3}$.
Two unknowns in exponents give a system of linear equations.
Find the value of the exponent $x$ so that the following equality holds: $2^x \cdot 2^3 = 2^{-1}$.
Find the value of the exponent $x$ so that the following equality holds: $2^x \cdot 2^3 = 2^{-1}$.
Solve the following exponential equation by expressing all terms as powers of the same base and operating with their exponents: $4^x \cdot 2^{x+1} = \frac{1}{8}$.
Solve the following exponential equation by expressing all terms as powers of the same base and operating with their exponents: $4^x \cdot 2^{x+1} = \frac{1}{8}$.
Find the values of $x$ and $y$ that simultaneously satisfy the following equalities, using the properties of powers to transform the system into a system of linear…
Find the values of $x$ and $y$ that simultaneously satisfy the following equalities, using the properties of powers to transform the system into a system of linear equations:
$2^x \cdot 4^y = 256$
$\frac{3^{2x}}{9^y} = 27$
Growth models
When a quantity is multiplied by the same growth factor $b$ in every time step, it follows
$$A = a \cdot b^{t}$$where $a$ is the starting amount and $t$ counts the steps. For a percent growth of $r$, the factor is $b = 1 + r$ (4% growth means $b = 1.04$). If the factor applies once every $p$ time units, use the exponent $t/p$.
Why? After 1 step you have $a \cdot b$. After 2 steps you multiply by $b$ again, giving $a\cdot b\cdot b = a \cdot b^2$, and so on. Repeated multiplication is exactly what an exponent counts.
Move the slider to see how the starting amount changes the curve $A = a \cdot 1.05^t$:
A culture of bacteria in a biology laboratory doubles every hour. If initially there are $1000$ bacteria, express as a product with a power the number of bacteria there…
A culture of bacteria in a biology laboratory doubles every hour. If initially there are $1000$ bacteria, express as a product with a power the number of bacteria there will be after $12$ hours and calculate its final value.
A principal is invested at an annual compound interest rate such that each year the money is multiplied by $1,05$. If the initial investment is $\$2000…
A principal is invested at an annual compound interest rate such that each year the money is multiplied by $1,05$. If the initial investment is $\$2000$, write the expression in the form of a power that calculates the balance after $t$ years. How much money will there be after $4$ years?
Moore's Law suggests that the number of transistors in a microprocessor doubles approximately every $2$ years. If in the year 1970 a processor had $2000$ transistors,…
Moore's Law suggests that the number of transistors in a microprocessor doubles approximately every $2$ years. If in the year 1970 a processor had $2000$ transistors, express the number of transistors in the year $1970 + t$ using a power with base $2$. Use this property to estimate how many transistors a processor would have in the year 2010.
Decay, inverse squares and huge numbers
If the factor is less than 1, the quantity shrinks. Dividing by 2 each step is the same as multiplying by $\frac{1}{2} = 2^{-1}$, and negative exponents are a natural way to write decay.
Some laws involve a power of the distance. For light or sound, the intensity is inversely proportional to the distance squared: if the distance is multiplied by $k$, the intensity is multiplied by $k^{-2} = \dfrac{1}{k^2}$.
For huge numbers, use the handy estimate $2^{10} = 1{,}024 \approx 10^3$. You can then convert between powers of 2 and powers of 10.
When a problem mixes units, convert before you write the equation. For example, $1$ km $= 1{,}000{,}000$ mm $= 10^6$ mm.
A radioactive substance reduces its mass to a third every day that passes. If today we have a sample of $81$ ounces, express using a power with base $3$ the amount of…
A radioactive substance reduces its mass to a third every day that passes. If today we have a sample of $81$ ounces, express using a power with base $3$ the amount of substance that will remain in $6$ days and calculate the result.
In astrophysics, the intensity of the light we receive from a star is inversely proportional to the square of the distance we are from it. If we move to a distance $4$…
In astrophysics, the intensity of the light we receive from a star is inversely proportional to the square of the distance we are from it. If we move to a distance $4$ times greater from the star, express using a power with a negative exponent the factor by which the original light intensity is multiplied. If the initial intensity was $I_0$, what will the new intensity be in terms of $I_0$?
Imagine you have a sheet of paper $0,1\text{ mm}$ thick. Every time you fold the sheet in half on itself, its thickness doubles. Express the thickness of the sheet after…
Imagine you have a sheet of paper $0,1\text{ mm}$ thick. Every time you fold the sheet in half on itself, its thickness doubles. Express the thickness of the sheet after $n$ folds using a power. If the average distance from the Earth to the Moon is approximately $384\,000\text{ km}$, set up an equation using powers of $2$ to estimate how many folds would be needed for the thickness of the paper to reach the Moon (you can estimate the value knowing that $2^{10} \approx 10^3$).