what is (2 + 1) (2^2 + 1) (2^4 + 1) … (2^2^10 + 1) + 1?
What Is (2 + 1) (2^2 + 1) (2^4 + 1) … (2^2^10 + 1) + 1?
Below are multiple fraction 91neg.coms capable of addition, subtraction, multiplication, division, simplification, & conversion between fractions và decimals. Fields above the solid black line represent the numerator, while fields below represent the denominator.
Use this 91neg.com if the numerators or denominators are very big integers.
+ - x /
= ?
In mathematics, a fraction is a number that represents a part of a whole. It consists of a numerator and a denominator. The numerator represents the number of equal parts of a whole, while the denominator is the total number of parts that biến hóa said whole. For example, in the fraction of
3
8
, the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore be
5
8
as shown in the image lớn the right. Cảnh báo that the denominator of a fraction cannot be 0, as it would make the fraction undefined. Fractions can undergo many different operations, some of which are mentioned below.
Addition:
Unlike adding & subtracting integers such as 2 & 8, fractions require a common denominator to undergo these operations. One method for finding a common denominator involves multiplying the numerators và denominators of all of the fractions involved by the hàng hóa of the denominators of each fraction. Multiplying all of the denominators ensures that the new denominator is certain to lớn be a multiple of each individual denominator. The numerators also need lớn be multiplied by the appropriate factors to lớn preserve the value of the fraction as a whole. This is arguably the simplest way to ensure that the fractions have a common denominator. However, in most cases, the solutions lớn these equations will not appear in simplified size (the provided 91neg.com computes the simplification automatically). Below is an example using this method.
a
b
+
c
d
=
a×d
b×d
+
c×b
d×b
=
ad + bc
bd
EX:
3
4
+
1
6
=
3×6
4×6
+
1×4
6×4
=
22
24
=
11
12
This process can be used for any number of fractions. Just multiply the numerators and denominators of each fraction in the problem by the product of the denominators of all the other fractions (not including its own respective denominator) in the problem.
EX:
1
4
+
1
6
+
1
2
=
1×6×2
4×6×2
+
1×4×2
6×4×2
+
1×4×6
2×4×6
=
12
48
+
8
48
+
24
48
=
44
48
=
11
12
An alternative method for finding a common denominator is to determine the least common multiple (LCM) for the denominators, then showroom or subtract the numerators as one would an integer. Using the least common multiple can be more efficient & is more likely to lớn result in a fraction in simplified form. In the example above, the denominators were 4, 6, and 2. The least common multiple is the first shared multiple of these three numbers.
Multiples of 2: 2, 4, 6, 8 10, 12
Multiples of 4: 4, 8, 12
Multiples of 6: 6, 12
The first multiple they all mô tả is 12, so this is the least common multiple. Khổng lồ complete an addition (or subtraction) problem, multiply the numerators và denominators of each fraction in the problem by whatever value will make the denominators 12, then add the numerators.
EX:
1
4
+
1
6
+
1
2
=
1×3
4×3
+
1×2
6×2
+
1×6
2×6
=
3
12
+
2
12
+
6
12
=
11
12
Subtraction:
Fraction subtraction is essentially the same as fraction addition. A common denominator is required for the operation to occur. Refer to the addition section as well as the equations below for clarification.
Multiplying fractions is fairly straightforward. Unlike adding & subtracting, it is not necessary lớn compute a common denominator in order lớn multiply fractions. Simply, the numerators và denominators of each fraction are multiplied, & the result forms a new numerator and denominator. If possible, the solution should be simplified. Refer khổng lồ the equations below for clarification.
a
b
×
c
d
=
ac
bd
EX:
3
4
×
1
6
=
3
24
=
1
8
Division:
The process for dividing fractions is similar to that for multiplying fractions. In order to divide fractions, the fraction in the numerator is multiplied by the reciprocal of the fraction in the denominator. The reciprocal of a number a is simply
1
a
. When a is a fraction, this essentially involves exchanging the position of the numerator và the denominator. The reciprocal of the fraction
3
4
would therefore be
4
3
. Refer to the equations below for clarification.
a
b
/
c
d
=
a
b
×
d
c
=
ad
bc
EX:
3
4
/
1
6
=
3
4
×
6
1
=
18
4
=
9
2
Simplification:
It is often easier to work with simplified fractions. As such, fraction solutions are commonly expressed in their simplified forms.
220
440
for example, is more cumbersome than
1
2
. The 91neg.com provided returns fraction inputs in both improper fraction form as well as mixed number form. In both cases, fractions are presented in their lowest forms by dividing both numerator & denominator by their greatest common factor.
Converting between fractions và decimals:
Converting from decimals lớn fractions is straightforward. It does, however, require the understanding that each decimal place lớn the right of the decimal point represents a nguồn of 10; the first decimal place being 101, the second 102, the third 103, and so on. Simply determine what power of 10 the decimal extends to, use that power nguồn of 10 as the denominator, enter each number to lớn the right of the decimal point as the numerator, and simplify. For example, looking at the number 0.1234, the number 4 is in the fourth decimal place, which constitutes 104, or 10,000. This would make the fraction
1234
10000
, which simplifies to lớn
617
5000
, since the greatest common factor between the numerator and denominator is 2.
Similarly, fractions with denominators that are powers of 10 (or can be converted to lớn powers of 10) can be translated lớn decimal size using the same principles. Take the fraction
1
2
for example. Lớn convert this fraction into a decimal, first convert it into the fraction of
5
10
. Knowing that the first decimal place represents 10-1,
5
10
can be converted lớn 0.5. If the fraction were instead
5
100
, the decimal would then be 0.05, & so on. Beyond this, converting fractions into decimals requires the operation of long division.
Common Engineering Fraction to Decimal Conversions
In engineering, fractions are widely used to describe the size of components such as pipes và bolts. The most common fractional và decimal equivalents are listed below.
