# Operations with Rational Numbers

Here we show you the operations with rational numbers.

## Sum of Rational Numbers

When adding and subtracting rational numbers, there are two different cases to consider. The first is when the two fractions have different denominators (the value located at the bottom of the fraction) and the second is when the two fractions have the same, or like denominators.
When adding and subtracting rational numbers with like denominators, the result is obtained by simply adding or subtracting the numerators (the top of the fraction) as follows:

65+35=6+35=95

When the denominators are not equal, a common denominator must be found. This is achieved by finding the least common multiple of the different denominators to create equivalent, while taking into account that any operation performed must also made to the numerator to not alter the value. For example, if the denominator is multiplied by 4 to find the least common multiple, the numerator must also be multiplied by 4.

14+65=520+2420=5+2420=2920

Note that the least common multiple of 4 and 5 is 20, so the first denominator must be multiplied by 5 and the second by 4 in order to obtain equivalent fractions with same denominator. Then the two numerators can simply be added as was shown in the previous operation. Read on about the operations with rational numbers.

## Multiplication of Rational Numbers

The multiplication of fractions is less complicated than addition or subtraction once the process is understood. First, the numerators of all the factors must be multiplied to give you the numerator of the product. Then the denominators can be multiplied regardless of whether the value is the same. This will result in the denominator of the product:

43×56×12=4×5×13×6×2=2036=1018=59

In this case the result could be simplified by dividing the numerator and denominator by the same number until both quotients can no longer be reduced.

In multiplication there is also an inverse element resulting in the number one, taking into account that the integers are also rational numbers and can be expressed as a fraction. To better explain, some examples are provided:

13×3=13×31=33=1

Similarly, among non-integer fractions, the same phenomenon also happens:

57×75=3535=1

## Division of Rational Numbers

Operations with Rational Numbers continued: When dividing rational numbers, the numerator of the first fraction is multiplied by the denominator of the second fraction and this will yield the numerator of the result. Then the denominator of the first fraction is multiplied by the numerator of the second fraction. This will yield the denominator of the result. Therefore in the case of division, the order of the ratios will alter the result. Consider the following example:

54÷23=5×34×2=158

As can be noticed, to divide rational numbers, one must use the process of cross multiplication, taking into account that the numerator and denominator of the first fraction do not change order, but those of the second fraction do change order to achieve the final result.

## Potentiation of Rational Numbers

For the potentiation of a rational number, the following operations with rational numbers are the rule:
If the rational number has different exponents for each and denominator, one must simply raise each base to the power of that exponent and simplify if possible:

anbm

2332=89

When you have the same number for the numerator and denominator, but different exponents for each, we can subtract the exponent of the denominator from the exponent of the numerator and simplify the fraction to an integer. For example:

aman=amn

3436=326=32

Although the same example can also be expressed proceed in this way:

3436=3×3×3×33×3×3×3×3×3=13×3=132=32

When raising rational numbers to a natural power, raise the numerator and denominator to that power:

(ab)n=anbn

(32)3=3323=278

In the event that the power is negative, we simply invert the fraction and raise it to that positive power:

(ab)n=(ba)n=bnan

(56)2=(65)2=6252=3625

If the power is -1, the fraction is simply reversed:

(ab)1=ba

(815)1=158

When the power is equal to 0, the result is 1:

(ab)0=1

(931)0=1

If the power is equal to 1, the result is the same rational number:

(ab)1=ab

(1743)1=1743

If powers with the same base are multiplied, the result remains the base and the exponents are added:

(ab)n×(ab)m=(ab)n+m

(34)2×(34)3=(34)2+3=3545=2431024

If we divide powers with the same base, we use the same principle as with the product, i.e. the base is maintained but the exponent of the second rational number is subtracted from that of the first:

(ab)n÷(ab)m=(ab)nm

(34)5÷(34)7=(34)57=3242=916

To solve the power of a power, the exponents must be multiplied:

[(ab)m]n=(ab)mn

[(23)3]2=(23)6=2636=64729

When multiplying different rational numbers with the same power, the fractions are multiplied and the exponent remains the same:

(ab)n×(cd)n=(a×cb×d)n

(23)2×(45)2=(2×43×5)2=(815)2

When dividing different rational numbers with the same power, the process of cross multiplication must be applied and the exponent remains the same:

(ab)n÷(cd)n=(a×db×c)n

(23)2÷(45)2=(2×53×4)2=(1012)2=(56)2

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