Unit 3 sec 1.5
powers
05 November 2015
22:05
As you know, ‘raising
a number to a power’ means multiplying the number by itself as specified number
of times. For example, raising 2 to the power of 3 gives
2³ = 2×2×2.
Here, the number 2 is called the base number or just base,
and the superscript 3 is called the power,
index or exponent. The word power is also used to refer to the result of
raising the number to a power - for example, we say that 2³ is a power of 2. When
we write expressions like 2³, we say that we are using index form or index notation.
This square and cube of a number of the results of
raising gates to the power of 2 and 3 , respectively. For example, the square 2
is 2² = 4, and the cube of 2 is 2³ = 8. The power 2 is read as ‘2 to the power
five’ or ‘2 to the five’. Other in dioceses are read in a similar way to 2.
Standard numbers
like billion and 8 trillion can be conventional described in index form. Some
basic rules were carrying out calculations with numbers written in index form.
It is worth getting to know the rules, as they will be useful later.
Multiplying numbers
in index form
sometimes you need to multiplying numbers in index form. For
example, suppose that you want to multiply 10² × 10³ = (10×10) × (10×10×10) =
10.
You can see that the total number of 10s multiplied together
is the sum of the indices, 2+3 = 5
To multiplying numbers in index form
that have the same base number, add the indices:
aⁿⁿ × aⁿ = aⁿⁿ+ⁿ.
Example 3
multiplying powers
is the following products
concisely in index form
(A)
3 × 3
(B)
5×5
(C)
2 × 3
(D)
2³ × 7 ×2² × 7²
(E)
9 × 3
Solution
(A)
3×3= 3+ =3 9
(B)
Multiplying by to
the power of 9 by increasing the index by 1, because the number 5 is the same
as 5¹
5×5 9 = 51+9= 5 to the power 10
(C)
©the product 2× 3 cannot be written any
more concisely, as the base numbers are different.
(D)
2³× 7 × 2²×7² = 2 3+2× 7 1+2 = 5 × 7³
(E)
the base numbers are different, but they can
be made the same.
9×3 = 3²×3= 3 2+5 = 3.
Dividing
numbers in index form
suppose that you wanted to by 10 by 10². You can do this
as follows:
10 ÷10² = 
= 
=
To divide numbers in index form that you have
the same base number subcontract the in indices.
Raising a number in
index form to a power
suppose that you want to find (10²)³, the queue of 10². To
raise any number to the power 3, you multiply 3 copies of the number together
stop so
(10²)³ = 10² × 10² × 10² = (10×10) × (10×10) × (10×10) =
Rising product pro
quotation to the power
there are 2 more facts that are often useful when you are
working with powers. Notice that
(2×10)³ = (2×10) × (2×10) × (2×10)
= 2×2×2×10×10×10
= 2³ ×10³.
There is an example of the 1st back. The 2nd
factor is similar, but it applies in motivations rather than the products
raising negative
numbers to powers
so far you have worked with natural numbers raised to
powers. Other numbers, such as negative numbers, can also be raised to powers.
For example,
(-2)² = (-2) × (-2) = 4
And
(-2)³ = (-2) × (-2) × (-2) = all × (-2)= -8
every pair of negative numbers multiplies together to give
a positive number. So you can see that
ª
negative number raised to an even power is positive
ª
negative number raised to an odd power is negative.
By
role that you can use when you are working with numbers in index form. These
rules apply to powers of any type of number, including negative numbers and
numbers that are not all. They are known as index laws, and they are summarised in the box below. You will meet
for other index laws later in the unit.
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