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Powers and roots

Operations with powers. Multiplication and division of powers.
Power of product of some factors. Power of a quotient (fraction).
Raising of power to a power. Operations with roots. Arithmetical
root. Root of product of some factors. Root of quotient (fraction).
Raising of root to a power. Proportional change of degrees of a
root and its radicand. Negative, zero and fractional exponents
of a power. About meaningless expressions.

 

 

Operations with powers.

 

1. At multiplying of powers with the same base their exponents are added:

 

a m ·  a n  =  a m + n .

 

2. At dividing of powers with the same base their exponents are subtracted:

 

 

3. A power of product of two or some factors is equal to a product of powers of these factors:

 

( abc ) n = a n · b n · c n

 

4. A power of a quotient (fraction) is equal to a quotient of powers of a dividend     (numerator) and a divisor (denominator):

 

( a / b ) n =  a n /  b n .

 

5. At raising of a power to a power their exponents are multiplied:

 

( a m ) n =  a m n .

 

All above mentioned formulas are read and executed in both  directions – from the left to the right and back.

E x a m p l e .  ( 2 · 3 · 5 / 15 ) 2 = 2 2 · 3 2 · 5 2  / 15 2  = 900 / 225 = 4 .

                                                                                                                          

Operations with roots.  In all below mentioned formulas a symbol  means an arithmetical root ( all radicands are considered here only positive ).

 

1. A root of product of some factors is equal to a product of roots of these factors:

 

2. A root of a quotient is equal to a quotient of roots  of a dividend and a divisor:

 

 

3. At raising a root to a power it is sufficient to raise a radicand to this power:

              

4. If to increase a degree of a root by n times and to raise simultaneously its    radicand to the n-th power, the root value doesn’t change:                                                                                            

                                                                         

 

5. If  to decrease a degree of a root by n times and to extract simultaneously the n-th degree root of the radicand,  the root value doesn’t    change:                                                                                              

  

                                                                                                                                      

Widening of the power notion. Till now we considered only natural exponents   of powers; but operations with powers and roots can result also to negative, zero and  fractional exponents. All these exponents of powers require to be defined.

 

Negative exponent of a power. A power of some number with a negative (integer) exponent is defined as unit divided by the power of  the same number with the exponent equal to an absolute value of the negative exponent:

 

 

Now the formula  a m : a n = a m - n may be used not only if  m is more than  n , but also for a case if  m  is less than  n .

 

E x a m p l e .  a4 :  a7 = a4 - 7 = a-3 .

 

If we want the formula   a m : a n = a m - n  to be valid at  m = n  we need the definition of zero exponent of a power.

 

Zero exponent of a power.  A power of any non-zero number with zero exponent is equal to 1.

   

E x a m p l e s . 2 0 = 1,   ( 5 ) 0 = 1,   ( 3 / 5 ) 0 = 1.     

 

Fractional exponent of a power.  To raise a real number  a  to a power with an exponent  m / n  it is necessary to extract the  n-th degree root  from the m-th power of this number a:

About meaningless expressions. There are some expressions:

 

Case 1.

 

    where  a 0 , doesn’t exist.                                                                                    

                                          

   Really, if to assume that   where x – some number, then according

   to the definition of a division we have:  a = 0 ·  x ,  i.e.  a = 0 ,  but this result

   contradicts to the condition:  a 0 .

 

Case 2.

 

    is any number.

 

  Really, if to assume that this expression is equal to some number  x , then

  according to the definition of a division:  0 = 0 ·  x . But this equality is valid

  at any number   x , which was to be proved.

 

Case 3.

            

If  to assume, that rules of operations with powers are spread to powers with

a zero base, then

 

   0 0   is any number .

 


 

S o l u t i o n . Consider the three main cases:

 

                       1)  x = 0 this value doesn’t satisfy the equation  ( Why ? ) ;

 

                       2)  at  x > 0 we receive: x / x = 1, i.e.1 = 1, hence, x – any number,

                            but  taking into consideration that in this case x > 0, the answer

                            is:  x > 0 ;

 

                       3)  at x < 0 we receive: – x / x = 1, i.e.  –1 = 1,  and the answer is:

                            there is no solution in this case.

 

                       So, the answer:  x > 0 .

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