Variables with Exponents – How to Multiply and Divide them?

In mathematics, a variable is a symbol which we use to represent an unknown quantity. We can depict a variable by symbols like x, y or z etc. When we multiply a variable by itself, the number of times we multiply it by itself is denoted by an exponent. There are certain rules for multiplying exponents and dividing them as well. Suppose we multiply z three times, so now z multiplied by itself three times is written as z3. So, 3 here is known as the exponent or the power of the variable z.

Exponential functions are a fundamental tool of algebra having its utility in many spheres of life, and students often struggle to understand it. But consistent practice and clarity of rules will help them in mastering the topic of exponents. To get more clarity on how to deal with exponents, one can take help of cuemath website. Cuemath has a dedicated team of teaching experts to guide the students through the fundamentals of algebra. Even for younger kids, one can get 3rd grade math worksheets which are very simple to practice in order to retain the conceptual knowledge.

Since the exponents signify repetitive multiplication. The basic structure of an exponent looks like x^y, where x is the base and y is its exponent. Or in other words, y is the number of times the variable x is multiplied by itself.

Before proceeding to the rules, there are some specific facts regarding the exponents like, the exponent is 1 when only the variable exists by itself, like z, in this case, it also means z1. And when the exponent is 0, that means the variable is not being multiplied by itself; hence the expression evaluates to 1, like zo=1.

Multiplication of Variables with Exponents

Suppose we have two variables like x2 and x4, so now how do we multiply them? Let us look at what does x2 and x4 exactly mean, x multiplied by itself twice, xx and x multiplied by itself four times which is xxxx so together if we count we see it has been multiplied 6 times.

Hence x2 x4 = (xx)(xxxx) = x 2+4 = x6

So just by adding the exponents, we get the answer.

Now what if we have mixed variables like x4 z or x3 y4 z.

In this case, we will add up the exponents of similar variables like x, y, z separately.

Hence if we multiply (x4 z) and (x z4), we add the exponents of x and z separately, hence (x4+1) (z1+4) = x5 z5

If constants are also present along with variables, we multiply the constants together and include it in the answer, like

Multiply 2x2 y and 3 x y2 , the answer is (2 x 3)( x 2+1) (y1+2) = 6x3 y3

Till now we have talked of only positive exponents, in case of negative exponents it implies division, like x-1 means 1/x, and y-2 = 1/y2.

Division of Variables with Exponents

Let us have a look at how to divide variables with exponents below.

So if the question is to solve x3/x2 then simplifying this expression it says (x x x) / (x x), and following the simple division method we would cancel all the common factors, and we will be left with just x. To make this look nicer, we can simply subtract the powers whenever division comes into the question so the answer would be x3-2 =x1 = x

If mixed variables are present, just subtract the powers of similar variables like (x4y3)/(xy2) = (x4-1)/(y3-2) = x3/y.

Multiplication and division of exponents are an essential aspect of algebra. It is because of these exponents that an algebraic expression is termed binomial or polynomial. Hence it is necessary to have clarity on them. Exponents also find their applications in the fields such as engineering, gaming, accounting, geography, economics, and finance etc.


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