Question

Determine the product of the following expression \[(8y+3)\times 4x\].

Answer 1

Hint: We will use the method of multiplication of variables to solve this question. So when variables are the same, multiplying them together compresses them into a single factor (variable). When multiplying variables, we multiply the coefficients and variables as usual. If the bases are the same, we can multiply the bases by merely adding their exponents.

Complete step-by-step answer:
Before proceeding with the question we should understand the concept of multiplication of variables.
If we are using two different variables, then all we have to do is write the two variables together without the multiplication sign. Example: \[x\times y=xy\] - this occurs because x and y represent two different amounts. Think of this in terms of calculating the area of a rectangle where the length and width are two different numbers but we do not know what those numbers are.
If we are multiplying a variable with itself then it is simply that variable squared or cubed or whatever power depending upon how many times you multiplied that variable by itself. Example: \[x\times x={{x}^{2}}\]. Thinking about this geometrically, again we think of this as calculating the area of a square whose side lengths are the same number but, like before, we don’t know what that number is.
 \[\,\Rightarrow (8y+3)\times 4x........(1)\]
As we know that \[x\times y=xy\] and any constant multiplied by the variable x is simply constant times the variable x. So applying these in equation (1) we get,
 \[\,\Rightarrow (4x\times 8y+4x\times 3).........(2)\]
Now simplifying and rearranging equation (2) we get,
 \[\,\Rightarrow (32xy+12x)\]
Hence the product is \[32xy+12x\].

Note: We have to remember the basic definition of multiplication of different variables and of multiplication of same variables to solve this type of question. We in a hurry may commit a mistake in multiplication of terms in equation (2) so we need to be careful with this.