Question

How do you distribute \[13x(3y+z)\]?

Answer 1

Hint: The given question is based on the distributive laws. It can be summarized as: whether you multiply say a number by a collection of numbers which are added together it will fetch you the same result as when you multiply the number with the individual numbers of the group and then add it.
For example - \[a\times (b+c)=(a\times b)+(a\times c)\]
In this example we have multiplied ‘a’ with the additional components present within the parenthesis i.e ‘b’ and ‘c’. So ‘a’ is multiplied by ‘b’ and ‘c’ separated by addition operator. Carrying out the addition operation first and then multiplying ‘a’ to it will yield the same result as multiplying ‘a’ with individual components ‘b’ and ‘c’ and then adding it. This is the distributive property.

Complete step by step answer:
According to the question we have to distribute \[13x(3y+z)\],
We will take this in two sections:
Firstly, we will multiply 13x with each of the added terms given in parenthesis and then add them
And thereby adhering to the distributive property.
 \[13x(3y+z)\]
\[\Rightarrow 13x(3y)+13x(z)\]
While carrying out multiplication care should be given as to which all numbers are present in an expression along with the variables used.
Multiplying each of the component individually we have,
\[\Rightarrow 39xy+13xz\]
We now have to add both the individual terms we got. But addition is only possible between two similar entities. The variables are different in this case, so it cannot be further added.
Hence we have the distributed form of the given question.
Therefore, the answer is \[39xy+13xz\].

Note:
While solving the question care should be given as to which property of the addition and multiplication is to be applied and whether it is associative, commutative or distributive property is to be determined.
1. Commutative property – It states that order in which numbers are multiplied or added does not matter. It gives the same result. That is, \[a+b=b+a\] or \[a\times b=b\times a\].
For eg – adding 2 and 3 can be done either ways:
\[2+3=5\]
\[3+2=5\]
\[\Rightarrow 2+3=3+2\]
2. Associative property- It states that in case of binary operations, the components in the parenthesis even on rearranging will give the same result. That is, \[(a+b)+c=a+(b+c)\]