Multiplying Exponents Calculator solves the problems of making calculations between two numbers with exponents. It takes the input of the two and their respective exponents and multiplies their exponents depending on the given scenario. You are not inclined to enter both of them in our Multiplying Exponents Calculator. If you only need to multiply the exponent of one number, it’s perfectly fine. Enter it and leave empty the second one. 

We will show you how to use the exponent calculator to multiply numbers with the same and different bases, negative exponents, and multiplying variables with exponents, without any problems.

If you want to check out more math-related calculators, click here.

What is an exponent?

I’m sure that all of you have come across this term before, but you may not still realize what it means and where it is used. If you are a newbie in math, or you never found interest in it, just by looking at the term, you would still claim that it is math-related.

History of the term “exponent”?

The term “exponent” originates and can be first seen in the late 16th Century in the Latin word “exponere“. Later “expound” was added to the English language dictionary from this Latin word, which by its definition means “to represent or explain something in detail”. At first glance, this term does not give you any valuable information for understanding its use in math. However, you will see later that it certainly has meaningful relations with the math concept.

Concept of exponents multiplication

Exponent is a math term that indicates how many times a base number is multiplied by itself. Exponents are always placed on top of the base they are attached to, on the right side. They give us this nice shortcut to write down how many times we should multiply a respective base by itself.

Instead of writing:

4 \times 4 \times 4 \times 4 = 256

We can easily write it as:

4^{4}

In some places, you may find a different term for it, power or index, but most importantly, they all mean the same and return the same result.

You need to be able to differentiate these two distinct math operations:

3 \times 3 is NOT the same as 3^{3}

How to multiply exponents? – Multiplication of exponents rules

We have already shown a few simple instances of the proper use of exponents in math. However, they are correct only if both a base and an exponent are positive. Furthermore, the provided ones were only for a single number, so let’s see what happens with two or more of them with exponents.

10^{2} \times 5^{3} => Base 10 is positive; Exponent 2 = also positive

That’s the most basic rule we use when multiplying exponents in math.

Let’s take a look at more complex multiplying exponents rules:

1. x^{0}= 1 – Every number raised to 0 equals 1.

2. x^{1}= x – Every number with an exponent of 1 will always return itself (unchanged).

3. x^{\frac{1}{2}} = \int x – Every number with an exponent of one-half equals its root.

4. x^{-x}= \frac{1}{x^{2}} – Every number raised to the negative power equals one over its power of two.

5. x^{m} \times x^{n} = x^{m + n} – Product of two numbers with the same base but raised to different power, we keep the same base and add their exponents.

6. \left (x^{m} \right )^{n}= x^{m \times n} – When we have a number with a double exponent, we keep the base and multiply the exponents

7. \left ( x \times y \right )^{m} = x^{m} \times y^{m} – Product of two numbers with the same exponent equals the multiplication of both numbers with their exponent, respectively

Multiplying exponents with the same base

Enough rules! Let’s roll up our sleeves and get hands dirty with some practice samples using the exponent’s calculator.

When you multiply numbers with the same base, keep them and add their exponents. Then you will have only one of them to the power of its exponent, which you can easily calculate.

a) Calculator example #1

Step:

X = 5
a = 2
Y= 5
b = 3

x^{a} \times y^{b} = 5^{5} = 25

b) Calculator example #2

Step:

X = 3
a = 0
Y= 3
b = 2

x^{a}\times y^{b} = 3^{3} = 9

c) Calculator example #3

Step:

X = 10
a = -1
Y= 10
b = 2

x^{a}\times y^{b} = 10^{1} = 10

Multiplying exponents with different bases

When you multiply numbers with different (not equal) bases and exponents, enter the values and let the calculator do it for you.

a) Calculator example #1

Step:

X = 5
a = 2
Y= 10
b = 3

x^{a}\times y^{b} = 25 \times 1000 = 25000

b) Calculator example #2

Step:

X = 1
a = 0
Y= 9
b = 2

x^{a}\times y^{b} = 1 \times 81 = 81

c) Calculator example #3

Step:

X = 4
a = 4
Y= 7
b = 3

x^{a}\times y^{b} = 256 \times 343 = 87808

Multiplying negative exponents

Let me show you how the multiplication of negative exponents works down below:

a) Calculator example #1

Step:

X = 2
a = -1
Y= 5
b = -2

x^{a} \times y^{b} = \frac{1}{2^{1}} \times \frac{1}{5^{2}} = 0.5 \times 0.004 = 0.002

b) Calculator example #2

Step:

X = 1
a = -1
Y= 4
b = -3

x^{a} \times y^{b} = \frac{1}{1^{1}} \times \frac{1}{4^{3}} = 1 \times 0.02 = 0.02

c) Calculator example #3

Step:

X = 5
a = 2
Y= 8
b = -2

x^{a} \times y^{b} = 25 \times 0.02 = 0.5

Multiplying variables with exponents

A variable is anything that we do not know its value because it changes often. Thus, we mark it mostly as “x” or “y”, meaning that the value is never constant. However, even if we don’t specify a base, we can still multiply the exponents. Let’s practice it:

a) Calculator example #1

Step:

X = X
a = 5
Y= X
b = -2

x^{a} \times y^{b} = x^{5} \times x^{-2} = x^{3}

b) Calculator example #2

Step:

X = Y
a = 2
Y= Y
b = 6

x^{a} \times y^{b} = y^{2} \times y^{6} = y^{8}

c) Calculator example #3

Step:

X = Z
a = 3
Y= Z
b = 3

x^{a} \times y^{b} = z^{3} \times z^{3} = z^{6}

We can use any letter for a variable definition, although “x”, “y” or “z” are the standard in math.