Radical Expressions and Equations


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Radical Expressions and Equations


The symbolis called the radical sign. It means root.

Example:



square root, the 2 is usually not written, it is implied

cube root

fourth root

fifth root







The number on the radical sign is called the index. It specifies the root.

acts as a grouping symbol.



4 means the square root of 4.

4 + 6 means the square root of four first, then add that to 6.

(4 + 6) means the square root of the sum of 4 and 6, that is, the square root of 10.

We can use a vinculum above the sum and remove the left and right parens like so:

It is common to say "the sum of 4 and 6 is under the radical sign."

The expression under the radical sign is called the radicand.

In the previous example (4 + 6) is the radicand.



At this point it will help to memorize the power table.



A root is a number which multiplied by itself index number of times gives the radicand as the result.

Example: 2 • 2 = 4, so 2 is the square root of 4 which is the radicand of √ 4 . (index is 2)

Example: 3 is the cube root of 27 since 3 • 3 • 3 = 27 which is the radicand of . (index is 3)



You will notice that (-2) • (-2) = 4 satisfies the definition of a root.        -2 is a square root of 4 as well.

The positive root is called the principle root. By convention, the radical sign by itself means the principle root.

If both roots are required then the plus-minus sign, ±, must be used.

Example: ±√ 16 is 4 and -4, which is writtem ±4



Even roots of a negative number will not be allowed for the Real domain. This restriction will be removed when we consider the Complex Number domain. Odd roots of negative numbers are fine.

(-4) is not allowed.     But is fine.   -2 is the cube root of -8. (-2)(-2)(-2) = (4)(-2) = -8

Example: is ± (-2) which is 2 and -2.       2 is not a root here since 2 • 2 • 2 • 2 • 2 = 32 not the radicand -32.

So the root here is just -2.


The radical sign is a fractional exponent.  Example, √ 4 is 4½ and is 81/3

Notice how the index becomes the denominator of the exponent. This will become useful later.



Since radicals are exponents then radicals obey the rules for exponents .



Simplifying and Evaluating Radicals




Multiplying radicals


Recall the power rule of exponents across factors.




(a b)n = an bn


 


Except when symplifying radicals we move from the right side of the equation to the left   like so:




anbn = (a b)n




For example:



using the product rule, a = 12, b = 3 and n = ½


 

Usually we are multiplying radicals with the same indices. But this does not need to be the case. When they are different we need to first rewrite the smaller index in terms of the larger index.


Example:


Note that 5 ½ = 5 ¼ + ¼ = 5 ¼5¼


so we can rewrite as


and we have which is


and finally:



Radicals with different indices with no common fators can be multiplied but you need to find the least common multiple of the indices involved.


Example:   

We need to the write the indices 2 and 3 in terms of 6.

Recalling that these indices are fractions we can proceed like so:


Radicals with fractional indices


Recall the power rule for exponents:





Example: (162)4 = 168


Example: (164)2 = 168


The index of the radical is by definition the fractional exponent of the radicand. The numerator of the fraction is the power of the radicand; the denominator of the fraction is as usual the index of the radical.


Example: x = x½

 

So   x3  means   (x 3 )½


and using the power rule for exponents this becomes


x3/2


Moving the other way,



Example:


In other words, the square root of a quantity squared is that quantity. This is true for any root.


Example: (we'll use the technique in this example in what follows)

 


another way to look at this is

 




Study the following carefully. The cummative law of multiplication is used on the exponents to write the exponential in two equivalent radical forms.



To summarize, the root of a power is the power of the root.



 

Moving factors to the outside of the radical


Factors of the radicand that have a power that is a multiple of the index can be brought outside the radical.



Example: √16 = √(42) = 4

Note: familiarize yourself with the radical sign as used here in its abbreviated form.


Example:



Example:


Here we need 16 in terms of a power of the index 3.     24 will not work.    But we notice that 24 can be written as (23)• 2 and we have the index matching the power of the factor (23).

 

Example:


The prime factors of 216 are helpful here. Let's use repeated division.


216/2 = 108

108/2 = 54

54/2 = 27

and 27 = 33


 So we must have


Of course, knowing that 63 = 216 makes this a bit faster.



Moving factors to the inside of the radical.


Here we proceed in reverse. Simply raise the factor to the index and write it inside the radical.


Example:


The index is 2, so we raise 4 to the second power and write it inside like so:





 

 


Example:


We have:





 



Dividing Radicals


We use the same rule for exponents as we did for multiplication noting the following:





Example:

The indices are the same, so we can combine under one radical sign.



Example:






Examples with different indices:


Strategy: rewrite radical factors so they have the same index so they can be combined then simplified.


The indices are 2 and 4.


8½ = 8¼ + ¼ = 8¼ 8¼


Now all radicals have the same index.



Combine under one radical sign. (division rule)


8/2 = 4



32 = 25 = 24 2



 

Radicals with algebraic expressions.


We assume that the radicand is positive for even indices. We are not allowing the square root of a negative number. For odd indices the radicand can be either positive or negative.


Example: is ok, but is not ok


is ok, as is


When algebraic expressions are the radicands then we limit the range of the independent variable in that expression to make sure the radicands are valid.


Examples:


x can not be negative here, x 0

x can be positive, negative, or zero since its square is positive (or zero).

x can not be negative here, x 0; if x were negative then x3   is less than 0 and we'd have the fourth root of a negative number which we are not allowing.

x can be be positive, negative, or zero

x can be zero or negative but not positive. If x were positive then -x is negative and the radicand would be negative.

The radicand must be greater than or equal to zero, so

x – 10 0, which limits x 10





Simplifying Radical expressions


Simplifying radical expressions means to move all possible factors outside the radicands and simplifying the result so that only one radical with extermal factors remains. Note, when simplified, a radical may not exist. Also, if radicals exist in denominators they must be moved to the numerator by a method called rationalizing the denominator.


Examples:

Here the radicand has a power higher than the index. So we must move the corresponding factor out.


The index is 2 so we must remove the highest power of each factor that has a power that is a multiple of 2. Here that factor is x2.


We can simplify no further.


The highest power of x that is a multiple of the index 2 is 4.

So we factor the radicand into x4 x


Next we write x4 in terms of the index (red 2).



Now we pull out the factor x2.


There is nothing more we can do.



Here's another example which is the same as the previous

but with a larger exponent on the radicand. Notice that the factor pulled out is a multiple of the index. That multiple is brought outside the radical. (The original power divided by the index.)


Example if the index were 4 and the factor had the power 12 then the resulting exponent of the factor pulled out would be 12/4 = 3.



Here we note that 16 = 24, but 4 is not a multiple of 3 but if we write this factor as 2322 then we have a factor with power 3, likewise for x5, which can be written as x3x2. So we rewrite the radicand in terms of these factors. and bring out the cubed factors.



Already simplified we have two terms, we need one. Note, this can be factored to give us (x+2)(x-2) to give us one term with two factors, but these factors are different.


Here we have one term with two factors 5 and x2. We can pull out the factor x2.


Here we have two terms but we can factor the 16 from these terms to get 16(x – 2), a term with two factors.


Now we have one term with a factor that is the 4th power of 2.


We can pull out the 2.




Our first step is to convert these three terms into one term. We recognize the square of the difference pattern and rewrite the radicand.


Now that we have a single term, we pull out the factor
(x – 2).


Notice,we do not need to constrain x since (x – 2)2 is always non negative.


Here we see x3 can be written as x2x, y4 as (y2)2




Write the single radical as the quotient of two radicals.


Bring out the factors and simplify.



x cannot be negative, that is x 0;  y can be any value.





Rationalizing the denominator


is not simplified. We need to remove the radical from the denominator. To do so we note that and multiply by this fraction.



Now recall that , so the denominator becomes 2 and the numerator becomes and we have:




Example:



We can write this as:


So we have


Now, the denominator names a cube root. That means we need to multiply this cube root by itself 3 times to remove the radical, so we need 2 more factors. The fraction we need is:



Now multiply


= =



Many times we need to use the special binomial product called the sum and difference to rationalize the denominator.


Recall that: (a + b)(a – b) = a2 - b2


So if we have this:

then to remove the radical we can multiply by

Now



So we have =



Example:








Rearrange the numerator to get the terms in the same order before multiplying.



We could have saved a bit of work if we had first noticed that the numerator is the negative of the denominator. Factor out -1 from either then cancel.


Like so:



This goes to show that it never hurts to take a second look before diving into the problem.



Example:



As usual we simplify first then try combining like terms.


Since 4 = 22, we can bring the 2 outside.



Now is common to both terms so we can factor it out to get


.





Solving Radical Equations


goal: Eliminate radicals from the equation to allow solving the equations by normal means.


strategy: Isolate one radical onto one side of the equation to raise the equation to the index of that radical so that the radical is eliminated and the equation can then be solved.


method: Manipulate the equation using addition/subtraction and multiplication/division to isolate the radical. Raise the equation to the power of the index. Repeat if multiple radicals exist in the equation until all radicals are eliminated.


check: Verify the answers in the original equation.



First we need the Principle of Powers:


For any positive integer n, if a = b then an= bn.


This principle allows us to "raise an equation" to any power.     However, by doing so we can introduce solutions that may not solve the original equation. We will see this effect as we proceed.


Example:



Here we have a single radical that is already isolated. The index is 2 so we square both sides of the equation.



x - 4 = 25


x = 29


check:



Example:



The radical is isolated and the index is 3. We raise the equation to the 3rd power, that is, we cube the equation.




2x – 3 = -27

 

2x = -24


x = -12





 

check:




Example:







add x to both sides to isolate the radical



now square both sides (index is 2) to eliminate the radical



move all terms to one side and simplify the equation



solve the resulting quadratic equation







we find that x can be -6 or 1



x = -6 does not solve the original equation.

x = 1 is the solution.


In the previous example we obtained two solutions, one of which was extraneous, x = -6.


Example:

This example has no Real solution. The index is even which means the root must be positive. Here the root is negative, that is, it is -6.



Example:



At least one radical is isolated, so can square both sides of the equation. We'll find that we'll still have the other radical, but we can isolate it then square the equation again to eliminate it, like so:



Now square both sides again.


m2- 28m + 196 = 16(m + 7)

 

m2– 28m + 196 = 16m + 112

 

m2 - 44m + 84 = 0


(m – 42)(m – 2) = 0


m = 42, m = 2


Check:



m = 42 is an extraneous root.



m = 1 is the solution to this equation.



Example:



First note that x cannot equal -2 since we cannot divide the 4 by zero.


Now clear the denominator by multiplying the equation by .


 



Now isolate the radical.



The index is 2, so square both sides. and solve the resulting quadratic equation.


9(x + 2) = 4 – 4x + x2


9x + 18 = 4 - 4x + x2


0 = -14 – 13x + x2


x2 - 13x -14 = 0


(x – 14) (x + 1) = 0


x = 14, x = -1


check:

Evaluate each side of the equation and check if they equal each other.



x =1 is a solution.



 




14 is an extraneous root.



x = 1 is the solution to this equation.




Example:






The radicals are isolated, but if we square both sides of the equation we're left with a radical on the right hand side. If we raise the equation to the 4th power then both radicals are eliminated.



We get x = 0, and x = -7/9 as possible answers


Checking x = 0, we find that the square root of 1 equals the fourth root of 1.


Checking -7/9, in the left radical we get the square root of a negative number,

so this is not a solution. (3(-7/9) + 1) = -7/3 + 1 = -4/3 < 0). -7/9 is an extraneous root.


The solution to this equation is 0.


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