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 m times and to raise simultaneously its
radicand to the
mth power, the root value doesn’t change:
5. If to decrease a degree of
a root by m times and to extract simultaneously the mth
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 .
a^{4} : a^{7} = a^{4
 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 nonzero 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
nth degree root from the mth
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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