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Below is a few practice problems of various difficulty, but you will need considerably more practice than one each. For that reason
you should definitely use the green “Try Another” button in the top right corner at least two or three times to complete
additional versions of these questions for more practice. You should keep using that button until doing these problems feels straight
forward and easy, and then come back after a week or so of doing other stuff and try again to make sure it is still just as easy for
you.
Theoretically Easier Difficulty Problem
Use your exponent properties to (correctly) combine the numerator and denominator. This can be done directly because they
are the same base.
To correctly combine the numerator and denominator use the property . In this case make sure to use parentheses around the
entire exponent from the top and the entire exponent from the bottom so that you can distribute the negative(s)
correctly.
Condense the following expression into a single exponential.
Split the exponent (correctly) to write this as a constant term, a term with x in the exponent and a term with y in the
exponent.
To correctly split the exponent, use the property
Expand the following exponential so that each exponent has at most one term.
Theoretically Medium Difficulty Problem
First try combining the numerator terms together using .
Next, make sure the numerator and denominator are using the same base. This likely means you will need to rewrite one base in
terms of the other base to some power. For example, you can rewrite as .
Don’t forget that you should rewrite the numerator in the denominator using (since you rewrote the exponent in the
denominator from the previous hint).
Condense the following expression into a single exponential.
Start by expanding out by term using .
Remember that you can break apart multiplication as well, using the property . Use this to get rid of the constant coefficients in
the exponents.
Expand the following exponential so that each exponent has at most one term.
Theoretically Harder Difficulty Problem
For a walkthrough of the harder version of this next problem, click the arrow to the right!
Condense the following expression into a single exponential.
(Hint: , )
Solution: Before we can do a lot, it helps to get everything to the same base. This is why a hint is provided in the situation
where they don’t all start the same base (Note: You shouldn’t always expect such a hint to be given though, so keep an eye out!)
The easiest way to do this is to simply replace the larger number with the universal base to the appropriate power in parentheses.
So in our case we will replace by and by and then simplify.
Step 1: Replace each base with universal base and power.
Step 2: Simplify power of power in each term.
Step 3: Distribute and Simplify.
Now that we everything in terms of the same base, we can begin merging. First we merge all the top bases together and all the
bottom bases together. Then when we are down to only one base with a (large and complicated) exponent, we will merge the top
and bottom bases together.
From above.
Product of bases equals sum of powers.
Simplify exponents.
Division of bases is subtraction of exponents.
Simplify Exponent.
Don’t get intimidated! This is just like the similar problem in the previous section - just harder! Start by getting all the bases
to be the same value - to do this pick the smallest base so that you can write all the other bases as a (positive) power of the base
you chose, then replace all those bigger bases with the correct power of that smaller base (in parentheses). For example, replace a
base of 8 with .
Now we want to combine all the bases in the numerator into one base with a large exponent, and similarly combine all the bases
in the denominator into one base with a large exponent.
Next combine the numerator and denominator bases together and simplify the exponents - but don’t forget to use parentheses to
make sure you distribute the negative correctly!
Condense the following expression into a single exponential.
(Hint: , , )
For a walkthrough of the harder version of this next problem, click the arrow to the right!
Expand the following exponential so that each exponent has at most one term.
Solution: Here we are essentially doing the reverse process of the last examples. Our goal is to expand out the expression by
writing the given single base as a product of bases with various powers. Moreover, the problems give you the expected bases. Thus
we will begin by separating the base on each addition symbol and then pull out the constant factor from each term to form the
different numeric bases.
Step 1: Separate terms as product of bases.
Step 2: Factor out largest constant from each exponent.
Step 3: Product of exponent is repeated power.
Step 4: Calculate bases.
We’ve done the majority of the work here to get the different bases that were expected (notice in the original problem we
wanted bases of , , and , which is exactly what we ended up with!) Now we need to simplify the exponents for each term by making
them negative if needed. Remember that the power doesn’t change magnitude, only the sign changes when you move a term from
the bottom to the top of a fraction (or from the top to the bottom).
From above.
Rewrite fractional exponents with negatives.
Rewrite to match original base order.
Don’t get intimidated! This is just like the similar problem in the previous section - just harder! Start by using negative
exponents to get the terms in the big exponent to not have complex fractions - for example you can rewrite a term in the exponent
like into which is a lot easier to look at and work with.
Split the exponent by expanding out (term by term in the exponent) using .
Remember that you can break apart multiplication as well, using the property . Use this to get rid of the constant coefficients in
the exponents.
Expand the following exponential so that each exponent has at most one term.
