Exponent of a Multiplication

$(5\times2)^3=$
Notice that we have a power of a product of whole numbers. We can see that the exponent $3$ is applied to the entire expression enclosed within the parentheses, therefore, we can raise each of the terms while maintaining the multiplication sign between them.
We will obtain:
$5^32^3=\times$
$125\times8=1000$

If we have a multiplication exercise with an exponent, we can apply the exponent to each of the terms separately.

$(2X)^3$

We will notice that the exponent applies to the entire expression enclosed in parentheses and that between the $2$ and the $X$ there is, in fact, a multiplication operation.

We can raise each of the terms of the expression to the exponent and we will obtain:

$2^3\cdot X^3=$

$8\cdot X^3=$

We will multiply the $X$ by its coefficient and we will get:

$8X^3$

$(2^2\cdot X^3)^2$

We see that the exponent $2$ is outside the parentheses, therefore, it applies to the entire expression. The terms of the expression are multiplied, therefore, it is the power of a multiplication.

Well, now we can apply the exponent to each of the terms separately and we will not forget to maintain the multiplication between them.

We obtain:

$\left(2^2\right)^2\cdot\left(x^3\right)^2$

We have obtained what is called: Power of a power

To be able to continue solving the exercise we will remember the following property:

Power of power

If we have a power of a power we must perform a multiplication of exponents, that is, multiply powers.

In symbols:

$(a^n)^k=a^{n\cdot k}$

First we will multiply the exponent located outside the parentheses by the exponent of the base $2$ and then, we will do it by the exponent of the base $X$.

We will get:

$2^4\cdot X^6$

$16\cdot X^6$

Now we will multiply the $X$ by its coefficient and we will have:

$16X^6$

which will demonstrate to us that no matter how many terms the exercise includes, as long as there is multiplication between them and as long as they are raised to a certain power located outside of the parentheses, in such a case we can apply the power to each of the terms separately and maintain the multiplication operation between them.

$(2^2X\cdot X^5\cdot X^2\cdot2\cdot5)^3$

Recommendation:

Before applying the power located outside of the parentheses to each of the terms separately, carefully observe the exercise.

If you look closely and are thoroughly familiar with the properties of powers or laws of exponents, you will immediately see that, you can first act inside the parentheses using the property of multiplication of powers with the same base to simplify the expression.

Let's remember that, when we have a multiplication of powers with the same base, we can add the exponents and obtain a single base with a single exponent.

If there is no exponent it means that the exponent is one and it is important that we remember to add it.

Upon observing the exercise we will discover that there are some equal bases: $2$ and $X$.

Since the operation between all the terms is multiplication we can add the relevant exponents and in this way we will arrive at:

$(2^3\cdot X^8\cdot5)^3=$

Now, with an already abbreviated expression between the parentheses, it will be easier and faster for us to apply the power located outside of the parentheses to each of the terms.

We will do it and obtain:

$\left(2^3\right)^3\cdot\left(x^8\right)^3\cdot\left(5\right)^3=$

$2^9\cdot X^{24}\cdot5^3=$

We can continue solving and we will arrive at:

$512\cdot X^{24}\cdot125=$

$64,000\cdot X^{24}=$

Let's multiply the $X$ by its coefficient and it will give us:

$64,000X^{24}=$

$(2\times5)^3=$

$(1\times2)^3=$

$(3\times3)^3=$

$(2\times2)^2=$

$(7\times4)^2=$

$\left(3^2X\times X^5\times X^3\right)^3=$

$\left(5^2X\times X^2\times X^2\right)^2=$

$\left(8^2X\times X^2\times X^25\times3X\right)^3=$

$\left(2^4\times5X^2\times X37\times3X\right)^3=$

$\left(2^4\times5X^2\times X\times8\times X^2\right)^3=$

How to solve a power of a multiplication?

To solve a power of a multiplication we must raise each of the factors to the indicated power and maintain the multiplication sign between the terms.

How to solve multiplication of powers with different bases?

The law of exponents indicates that when we have multiplication of powers with the same base, we must add the exponents, however, if we have different bases we cannot apply this law.

How to multiply powers with different bases and the same exponent?

If the base is different we cannot add the exponents, although in some exercises it will be possible to manipulate the expressions to equalize the bases and apply the law of exponents.

How to solve addition of powers with different bases?

There is no law for the addition of powers. It does not matter if they have the same base or not.

How are powers multiplied with the same exponent and different bases?

If the base is different, we cannot directly apply the law of exponents. Sometimes we may manipulate the bases to try to equalize them and apply the law.

If you are interested in this article, you might also be interested in the following articles:

• Powers
• The Rules of Exponentiation
• Division of Powers of the Same Base
• Power of a Quotient
• Power of a Power
• Power with Zero Exponent
• Powers with a Negative Integer Exponent
• Taking Advantage of All the Properties of Powers or Laws of Exponents
• Exponentiation of Whole Numbers
• Multiplicative Inverse
• The Multiplication Tables

In the Tutorela blog, you will find a variety of articles about mathematics.