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Simplifying Radical Expressions
Learning Objective(s)
·
Simplify numeric and algebraic radical expressions.
Introduction
Radical expressions
are expressions that include a
radical, which is the symbol for taking a
root. They come in many forms, from simple and familiar, like
_{
}, to quite complicated, as in
_{
}. In any case, we can use what we know about
exponents
to make sense of such expressions.
Let’s start by exploring radicals; we’ll worry about how to simplify them later on.
The Radical
A radical is a mathematical symbol used to represent the root of a number. Here’s a quick example: the phrase “the square root of 81” is represented by the radical expression
_{
}. (In the case of square roots, this expression is commonly shortened to
_{
}—notice the absence of the small “2.”) When we find
_{
} we are finding the nonnegative number
r
such that
_{
}, which is 9.
While square roots are probably the most common radical, we can also find the third root, the fifth root, the 10^{th}
root, or really any other
nth root of a number. The
nth root of a number can be represented by the radical expression_{
}.
Radicals and exponents are inverse operations. For example, we know that 9^{2}
= 81 and
_{
} = 9. This property can be generalized to all radicals and exponents as well: for any number,
x, raised to an exponent
n
to produce the number
y, the
nth root of
y
is
x.
We can represent this property like this:
_{
}. A warning though: it is always true if
x
≥
0, and it is always true if
n
is odd. But it is not true when
x
< 0 and
n
is even.
Why is this the case? It is because raising any number, positive or negative, to an even power has the effect of making the new number positive. This is not the case for odd exponents. For example, think about inserting
x
= 3 and
n
= 2 into the formula above.
The radical would be written as
_{
}, which works out to_{
}, or 3. But our initial
x
value was 3, so we are left with the statement 3 = 3. This is not true!
Pulling an exponent out of a radical
When working with exponents and radicals:
·
If
n
is odd,
_{
}.
·
If
n
is even,
_{
}. (The absolute value accounts for the fact that if
x
is negative and raised to an even power, that number will be positive, as will the
nth root of that number.)
Note that this gives us two cases for when
n
is even:
·
If
x
³
0 and
n
is even,
_{
}.
·
If
x
< 0 and
n
is even,
_{
}.
Working with Radicals
Taking the square root of a number requires us to break down, or factor numbers. We need to find out what number can be multiplied by itself to produce the number we have.
If we were asked to find_{
}, for example, the thought that 16 = 4^{2}
= 4
•
4 would probably come to mind. Hey now—we just factored 16 into 4
•
4.
Factoring lies at the root of simplifying radical expressions. If we understand exponents as repeated multiplication, then we can think about radicals and roots in the same way—although the way we think about the repeated multiplication under the radical sign may be a bit different than what we are used to.
Let’s explore this idea of factoring using the radical expression_{ }. We can read this as “the third root of 125” or “the cube root of 125.” To simplify this expression, we are looking for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. Let’s factor 125 and see if we can find that number.



_{ } 

_{ } 
125 ends in 5, so we know that 5 is a factor. We expand 125 into 5 

_{ } 
We factor 25 into 5 and 5. 

_{ } 
We found the factors: 5 


5 
The prime factors of 125 are 5
•
5
•
5, which can be rewritten as 5^{3}. The cube root of a cubed number is the number itself, so
_{
}. We have found the cube root, the three identical factors that equal 125.
125 is known as a perfect cube. This means that the cube root of 125 is an integer.
Similarly, 81, 64, and 49 are
perfect squares, because their square roots are also integers (9, 8, and 7, respectively).
All
even
roots (square roots, fourth roots, sixth roots, etc.) are positive numbers. So, for example,
_{
} and
_{
} must be positive. This means then, that even roots exist only for positive numbers. A radical like_{
}
is impossible to evaluate, as no number can be multiplied by itself to produce 25.
Odd
roots (third roots, fifth roots, seventh roots, etc.) are a different story, though. We can find an odd root of a negative number, such as
_{
}. This radical expression simplifies to 2 because 2
•
2
•
2 = 8.
Find the square root of 324.
A) 16
B) 18
C) 21
D) 162
Show/Hide Answer
A) Incorrect. 16^{2}
= 256. To find the square root of 324, factor 324 and look for pairs of common factors. If you factor 324, you will find that 324 = 2
•
2
•
9
•
9, or 18
•
18. The number 324 can also be written as 18^{2}. The correct answer is 18.
B) Correct.
_{
}
= 18. If you factor 324, you will find that 324 = 2
•
2
•
9
•
9, or 18
•
18. The number 324 can also be written as 18^{2}.
C) Incorrect. 21^{2}
= 441. To find the square root of 324, factor 324 and look for pairs of common factors. If you factor 324, you will find that 324 = 2
•
2
•
9
•
9, or 18
•
18. The number 324 can also be written as 18^{2}. The correct answer is 18.
D) Incorrect. 162^{2}
= 26,244. Dividing 324 by 2 will not result in the number’s square root; try factoring 324 and looking for pairs of common factors. If you factor 324, you will find that 324 = 2
•
2
•
9
•
9, or 18
•
18. The number 324 can also be written as 18^{2}. The correct answer is 18.
Now let’s look at a radical that is not a perfect square root:
_{
}. We can find the root of this radical by using the same method as we used for
_{
}. We will factor the number under the radical (also known as the
radicand), 63, looking for pairs of factors that can be expressed as a power.



_{ } 


Factor 63 into 7 and 9. 


Factor 9 further into 3 and 3. 


Rewrite 3 


Separate the radical into the product of two factors, each under a radical. 


Take the square root of 3^{2} 



Rearrange terms 
So
_{
} is another way of writing_{
}. We used factoring as well as the idea that
_{
} to simplify this radical. We also used another useful trick—we pulled the factors under a radical apart into separate factors, each under its own radical. This trick is called the Multiplication Property of Square Roots. It lets us pull out the perfect squares from the factors that cannot be simplified further.
Simplest Form
Radicals will sometimes include variables as well, as the expression
_{
}. To simplify these radicals we still use factoring, but we’ll have to apply the rules of exponents, too. Let’s try it:






Separate terms, look for squared numbers and variables. Factor 49 into 7 


Factor 


Separate the squared terms into individual radical terms 

7 
Take the square roots of each radical term. We do not know whether 

7xy 
Combine like terms and simplify 


7xy^{2} 
We find that the square root of 49x
^{2}
y
^{4}
is 7xy
^{2}. In order to check this calculation, we could square 7xy
^{2}, hoping to arrive at 49x
^{2}
y
^{4}. (And, in fact, you would get this expression if you evaluated (7xy
^{2})^{2}.)
7xy
^{2}
is also known as the simplest form of this radical expression.
Simplest Form
To be in simplest form, a radical expression:
·
Has been factored completely and all perfect factors removed;
·
Cannot contain a fraction;
·
If in fractional form, the denominator cannot contain a radical expression.
Simplest form is not necessarily a “better” way of representing a radical expression, it is just one way to do it. In fact, some problems may be easier to solve with a radical expression that has not been simplified. However, using simplest form can help to make sense of more complicated radical expressions.
Let’s simplify one last expression that includes variables and fractions:






Factor the coefficient 24 into 2 


Factor the variables. We are looking for cube exponents, so we factor 


Separate the terms into individual radical terms 


Simplify, using the property that 


Combine like terms 



The simplest form of this expression. There are no radicals in the denominator, no fractions in the radical, and all cubes have been pulled out of the radical expression. 
Whew! That was complicated, but by taking it one step at a time, we found the solution.
Simplify
_{
}
.
A)
_{
}
B)
_{
}
C)
_{
}
D)
_{
}
Show/Hide Answer
A) Incorrect. 8h
^{3}
is the square root of 64h
^{6}, not the cube root. The answer is 4h
^{2}.
B) Incorrect. To find the cube root of 64h
^{6}, find the cube root of 64, and then the cube root of
h
^{6}. The answer is 4h
^{2}.
C) Incorrect. 4 is the cube root of 64, but +
h
does not factor into the expression. The answer is 4h
^{2}.
D) Correct. To simplify the expression, find the cube root of 64 and the cube root of
h
^{6}.
Summary
A radical expression is a mathematical way of representing the
nth root of a number. To simplify radical expressions, look for exponential terms within the radical, and then use the property
_{
} to pull out quantities. But keep in mind that while this property is always true if
x
≥
0, and it is always true if
n
is odd, different things start to happen when
x
< 0 or
n
is even. All rules of exponents apply when simplifying radical expressions.
Combine and Simplify These Radicals.
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