RADICAL EXPRESSIONS
Math 2
Solve the following:
1.) Use a 2  b 2  c 2
2.) x 2  49
3.) 500  5k 2
c
6
8
In each case above, what was the last step you used in solving your work?
In your work, what did the taking the square root accomplish?
4.) Let’s say you invest $500 into an account. Your goal is to have $600 in that account two years from
now. What kind of an interest rate will you need to get from the bank in order to do this?
Using the investment formula A  P0 (1  r )t , where A equals the amount of money in the bank (at
any given time), P0 is the Principal amount of money at time zero (your initial investment),
r is the annual interest rate (the percent the account gives you), and t is the time (in years), we
could solve the equation …
600  500(1  r )2
Start by dividing each side by 500.
Then take the square root of each side.
(to cancel out the square)
Finally, subtract 1 from each side.
Approximately what interest rate will you need to get for the account to reach $600 in two years?
Verify your answer by substituting the interest rate back into the equation & checking your solution.
5.) Let’s say you invest $500 into an account. Your goal is to have $1000 in that account three years
from now. What kind of an interest rate will you need to get from the bank in order to do this?
We could solve the equation …
1000  500(1  x)3
Start by dividing both sides by 500.
Does taking the square root of both sides make sense now? Why not?
To cancel out a cube, such as what we have in this equation, we need to take a cube root. A cube root
looks like this…
3
1
power.
3
1
Go back to where we left off in this equation and take the cube root of both sides (raise to the ).
3
and can be typed in on your calculator by raising a base to the
Finally, subtract 1 from each side.
Approximately what interest rate will you need to get for the account to reach $1000 in three years?
Verify your answer by substituting the interest rate back into the equation & checking your solution.
-----------------------------------------------------------------------------------------------------------------------------------The volume of a cube is calculated by raising the length of a side to the power of 3.
The formula is V  s 3 , where V is the volume (in units cubed) and s is the side length.
6.) What is the volume when the side length is 3 inches?
s
s
s
7.) What is the side length when the volume is 3375 cm 3 ?
8.) What is the side length when the volume is 246 cubic feet? Round to two decimal places.
9.) Let’s say you invest $500 into an account. Your goal is to have $1000 in that account four years
from now. What kind of an interest rate will you need to get from the bank in order to do this?
We could solve the equation …
1000  500(1  x)4
Follow the same steps as you did in problem #5. However, instead of using a cube root, you will need to use a
4
th
and can be typed in on your calculator by raising a base to the
4 root, which looks like this:
1
4
power.
Solve the equation:
1000  500(1  x)4
Approximately what interest rate will you need to get for the account to reach $1000 in four years?
Verify your answer by substituting the interest rate back into the equation & checking your solution.
-----------------------------------------------------------------------------------------------------------------------------------------------To solve an equation that involves a variable raised to an integer exponent involves using different roots.
For example, you use a square root to cancel out a square, a cube root to cancel out a cube, a fourth root to
cancel out a fourth power, etc.
x2
The symbol
3
x3
4
x4
18
x18
n
xn
is called the RADICAL symbol. The number located in the “hitch” is called the INDEX.
Solve the following equations using radicals.
10.) x5  1024
11.) x 7  279936
12.) 6000  2000(1  x)12
RADICALS AND RATIONAL EXPONENTS
n
From our previous work, we’ve seen that radicals,
, can be used to solve equations with integer exponents.
1
1
We’ve also seen that expressions like 3 x can be re-written as x 3 and 7 x can be re-written as x 7 .
When we write exponents as fractions, we call these RATIONAL EXPONENTS (see previous notes on ‘rational’).
To summarize:
n
x  __________
1
m
x  __________
~ and ~
Calculate the following:
1
1
1
1.) 49 2 
3.) 9 2 
2.) 100 2 
1
2
x  __________
Summarize:
Did you notice a pattern?
Note: Writing an index of 2 (number in the “hitch”) is optional when writing the square root. In other words,
x
means the same thing as
2
x , and vice versa.
-----------------------------------------------------------------------------------------------------------------------------------------------Simplify the following.
 
3.) x 2
3
3
 _________
 13 
6.)  x   _________
 
 
4.) x 4
8
 _________
 
5.) x 7
5
 15 
7.)  x   _________
 
We previously learned for expressions such as
x 
a b
 
8.) x7
2
1
7
 _________
 _________
 x ab .
If we apply that to rational exponents, as in the cases of problems 6, 7, and 8, we can summarize:
1 n
 n
1
x   x  x
 
Previously, we explored the formula A  P0 (1  r )t , where A equals the amount of money in the bank
(at any given time), P0 is the Principal amount of money at time zero (your initial investment),
r is the annual interest rate (the percent the account gives you), and t is the time (in years).
However, for some investments, the interest rate is given for some period of time other than
annually.
Consider the following. You are looking to invest $500, but you are told the account gains 22.5%
interest over three years.
We can modify the formula for interest as follows:
A  500 1.225
t
3
where $500 is the initial investment, 1.225 is the growth rate, calculated by adding 1 + .225, and
t
3
represents the time (in years) divided by the 3 years it took to reach 22.5%.
9.) Substitute 3 in for t to find the amount of money you will have in the account at the end of 3
years.
Next, you would like to figure out the approximate annual interest rate. The initial investment of $500 does
not affect the annual interest rate. In trying to figure out the annual interest rate, should we…
1
10.) Figure out what 1/3 of 1.225 is?
Figure out what 1/3 of .225 is?
Figure out what (1.225) 3 is?
Calculate each of the above. Then we will discuss which one must be correct in the context of this problem.
To see which one is correct, substitute your answers from the above questions into the following annual
interest formulas. Then use a calculator to determine which of these is correct.
11.)
A  500(_______)3
A  ______________
A  500(_______)3
A  500(_______)3
A  ______________
A  ______________
One of these should (closely) match your answer from #9.
12.) After your calculations, what is the correct annual interest rate? ________________
Strontium-90 has a half-life of 29 years. The half-life formula for its rate of decay is:
A  P0 (0.5)
t
29
where A represents the amount of remaining Strontium-90, P0 represents the initial amount of Strontium-90,
and t is the time (in years).
13.) Calculate the annual rate of decay (Remember, decay rate = 1 – r).
14.) Rewrite the formula for Strontium-90 as annual exponential decay.
Now compare the two formulas. Let’s say the initial amount of Strontium-90 is 1000 mg.
15.) Use the half-life formula to calculate the amount of Strontium-90 that will remain after 29 years.
16.) Use your formula from #14 to calculate the amount of Strontium-90 that will remain after 29 years.
Did you get (approximately) the same answers for #15 & #16? If not, check with your teacher.
17.) Use both formulas to calculate the amount Strontium-90 that will remain after 1000 years.
Did you get (approximately) the same answers for both formulas? If not, check with your teacher.