Table of Contents |
Not all numbers have a nice even square root. For example, if we found on our calculator, the answer would be 2.828427124746190097603377448419... and even this number is a rounded approximation of the square root. Decimal approximations are good estimates, but you may need the exact value. To do this, we will simplify radical expressions using the product property of square roots:
We can use the product rule to simplify an expression such as by splitting it into two roots, and simplifying the first root to get . The trick in this process is being able to translate a problem like into . There are several ways this can be done.
The most common and, with a bit of practice, the fastest method, is to find perfect squares that divide evenly into the radicand, or number under the radical.
EXAMPLE
75 is divisible by 25, a perfect square | |
Split into factors | |
Product rule, take the square root of 25 | |
Our Solution |
If there is a coefficient in front of the radical to begin with, the problem merely becomes a big multiplication problem.
EXAMPLE
63 is divisible by 9, a perfect square | |
Split into factors | |
Product rule, take the square root of 9 | |
Multiply coefficients | |
Our Solution |
As we simplify radicals using this method it is important to be sure our final answer can be simplified no more.
EXAMPLE
72 is divisible by 9, a perfect square | |
Split into factors | |
Product rule, take the square root of 9 | |
But 8 is also divisible by a perfect square, 4 | |
Split into factors | |
Product rule, take the square root of 4 | |
Multiply |
The previous example could have been done in fewer steps if we had noticed that , but often the time it takes to discover the larger perfect square is more than it would take to simplify in several steps.
We can simplify higher roots in much the same way we simplified square roots, using the product property of radicals.
Often we are not as familiar with higher powers as we are with squares. It is important to remember what index we are working with as we try and work our way to the solution.
EXAMPLE
We are working with a cubed root, want third powers | |
Test 2, , 54 is not divisible by 8 | |
Test 3, , 54 is divisible by 27! | |
Write as factors | |
Product rule, take cubed root of 27 | |
Our Solution |
Just as with square roots, if we have a coefficient, we multiply the new coefficients together.
EXAMPLE
We are working with a fourth root, want fourth powers | |
Test 2, , 48 is divisible by 16! | |
Write as factors | |
Product rule, take fourth root of 16 | |
Multiply coefficients | |
Our Solution |
Now let's talk about the quotient property of radicals, which states that if you have the nth root of and you divide that by the nth root of b, you can write this as a single quotient, , and then take the nth root of that. Notice again that our roots must be the same.
EXAMPLE
Use quotient property of radicals | |
Divide 18 by 2 | |
Simplify | |
Our Solution |
Let's try another example going in the other direction.
EXAMPLE
Use quotient property of radicals | |
Since 27 and 8 are both perfect cubes, take the cubed root of the numerator and denominator | |
Our Solution |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License