3.01  Introduction to  Radicals

Intermediate Algebra: One Step at a Time.  Page 243 - 248:   #31, 32, 33, 44, 53, 59

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

P. 246, # 31.  

Solution:  Find a perfect square that divides into .  That would be  times .  Place the  in the first radical and the in the second radical.

Since this is a numerical problem you can check the answer by calculating the decimal value of the problem and then calculating the value of the answer to see if these values are the same.

                       = 6.708203932

                    = 6.708203932 --  It checks!!

P. 248, # 32.  

Solution:  Before you begin cube root problem, you must remember the perfect cube numbers, i.e., , and find the perfect cube that divides evenly into . 

The only perfect cube that divides evenly into  is .  Break down the  into .

Since this is a numerical problem you can check the answer by calculating the decimal value of the problem and then calculating the value of the answer to see if these values are the same.

                          = 3.77976315

                    = 3.77976315  --  It checks!!

P. 248, # 33.  

Solution:  Before you begin cube root problem, you must remember the perfect cube numbers, i.e., , and find the perfect cube that divides evenly into . 

The only perfect cube that divides evenly into  is .  Break down the  into .

Since this is a numerical problem you can check the answer by calculating the decimal value of the problem and then calculating the value of the answer to see if these values are the same.

                           = 4.30886938

                    = 4.30886938  --  It checks!!

Preliminary thoughts to precede #38 - 64:  Do you remember the law of exponents about raising a power to a power?  When you raise a power to a power, you must multiply the exponents!  Taking a root of a power is the opposite of raising a power to a power, so you must divide the exponent by the index of the radical.  By the way, the index of a square root is 2, the index of a cube root is 3, the index of a fourth root is 4, etc.  It is a very good thing when the index of the radical divides evenly into the exponent.  A few simple examples might help.

;                ;              ;              ;             ;   

;               ;            ;              ;              .

If the index does not divide evenly into the exponent, then find an exponent that IS divisible by the index, and break it down as illustrated by the following examples.

P. 246, # 44.     

Solution:    When taking a fourth root of variables raised to powers, you must divide the exponents by 4.  In this case, the exponents are NOT divisible by 4, so find a number that IS divisible by 4, and break it down into two radicals as follows:

Find a power that IS divisible by 4 and place it in the first radical.

Place all the “leftover” factors in the second radical:

Simplify the first radical by the dividing exponents by the index.  The second radical can’t be simplified, so leave it alone.

P. 247, # 53.  

Solution:  Find a perfect square that divides into 98.  That would be 49 times 2.  Place the 49 in the first radical, along with powers of x and y that are divisible by 2.

P. 248, # 59.  

Solution:     When taking a fourth root, you must first find a perfect fourth power factor (i.e., ) that divides into 32.  For the powers of x and y, you need to find exponents that are divisible by 4.  In this case, the power of x is divisible by 4 so goes in the first radical with the perfect powers.  Break down the  into .

The power of y must be broken into .  Place the perfect fourth power  in the first radical, and the leftover factor in the second radical:

Simplify the first radical by taking the fourth root of 16, which is 2, and divide the exponents by the index of the radical which is 4.  The second radical can’t be simplified, so leave it alone.

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