Question

How do you simplify $3(6m + 3) + 8(m + 6) + 9(4m + 5)$ ?

Answer 1

Hint: In this question, we need to simplify the given polynomial. Firstly, for the first term we will multiply 3 to $(6m + 3) $. Then we simplify the second term by multiplying 8 to $(m + 6) $. After that we simplify the third term by multiplying 9 to $(4m + 5) $. We do this multiplication using the distributive property of addition. We then combine all the expressions obtained and simplify. After that we combine the like terms and obtain the required algebraic expression.

Complete step by step answer:
Given an algebraic expression of the form,
$3(6m + 3) + 8(m + 6) + 9(4m + 5) $ …… (1)
We are asked to simplify the expression given in the equation (1).
We simplify the given expression term by term and the combine all together to obtain the simplified form.
Here we make use of distributive property of addition for each of the terms in the given expression.
According to the distributive property of addition, sum of two numbers multiplied by the third number is equal to the sum of each number multiplied by the third number.
For instance, $a \cdot (b + c) = a \cdot b + b \cdot c$ …… (2)
Now let us consider the first term given as, $3(6m + 3)$
Here $a = 3, $ $b = 6m,$ $c = 3$
Using distributive property of addition given in the equation (2), we get,
$3(6m + 3) = 3 \cdot 6m + 3 \cdot 3$
$ \Rightarrow 3(6m + 3) = 18m + 9$
Now let us consider the second term given as, $8(m + 6)$
Here $a = 8, $ $b = m,$ $c = 6$
Using distributive property of addition given in the equation (2), we get,
$8(m + 6) = 8 \cdot m + 8 \cdot 6$
$ \Rightarrow 8(m + 6) = 8m + 48$
Now let us consider the third term given as, $9(4m + 5)$
Here $a = 9, $ $b = 4m,$ $c = 5$
Using distributive property of addition given in the equation (2), we get,
$9(4m + 5) = 9 \cdot 4m + 9 \cdot 5$
$ \Rightarrow 9(4m + 5) = 36m + 45$
Now substituting all the obtained expressions in the equation (1) and simplifying we will get the desired answer.
Hence from equation (1), we get,
$ \Rightarrow (18m + 9) + (8m + 48) + (36m + 45)$
Now writing without the parenthesis, we get,
$ \Rightarrow 18m + 9 + 8m + 48 + 36m + 45$
Rearranging the above expression we get,
$ \Rightarrow 18m + 8m + 36m + 9 + 48 + 45$
Combining the like terms $18m + 8m + 36m = 62m$
Combining the like terms $9 + 48 + 45 = 102$
Hence we get,
$ \Rightarrow 62m + 102$

Hence the simplified form of the algebraic expression $3(6m + 3) + 8(m + 6) + 9(4m + 5)$
is given by $62m + 102$.

Note: The distributive property applies to the multiplication of a number with the sum or difference of two numbers, i.e. this property holds true for multiplication over addition and subtraction. It simply states that multiplication is distributed over addition or subtraction.
Let a, b, c be any real numbers.
The distributive property of addition is given by,
$a \cdot (b + c) = a \cdot b + b \cdot c$
The distributive property of subtraction is given by,
$a \cdot (b - c) = a \cdot b - b \cdot c$
Also we must know which mathematical expressions have to be used to simplify the equation.