### Example of Multiplication of Powers with the Same Base

$5^3\times5^{-2}\times5^5=$ Since the bases are the same we can add the exponents.

** Then, we will apply the new exponent (result of the addition) to the base:**

$5^{3+(-2)+5}=$

$5^6=15625$

## Examples of multiplying exponents with the same base

**If we realize that in a certain exercise, terms with the same bases are multiplied, we can add their exponents and apply the new exponent obtained to the base.**

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### Let's look at other examples

$x^3\cdot x^4+4^2\cdot4=$

**In this exercise, we see** **$2$**** different bases,** **$X$**** and** **$4$****.**

Notice that between the $X$s there are multiplication signs. According to the property of powers with the same base, we can add the exponents of the $X$s, obtain a new exponent, and apply it to the $X$.

**We will do it and obtain:**

$X^7+4^2\cdot4=$

Now let's see that we can also add the exponents that have base $4$ and obtain a single exponent that we can apply to that number.

Attention: if there is no exponent, it means that the exponent is $1$.

**We will do it and obtain:**

$X^7+4^3=$

### Now let's look at a slightly more complicated exercise

$4\cdot X^2\cdot X^3-2\cdot X^5=$

Let's not panic, we will work according to the order of mathematical operations.

Let's pay attention to the first part before the subtraction sign. We have terms with the same base ($X$) and, among them, the multiplication sign.

**We can add the exponents and obtain the following expression:**

$4\cdot X^5-2\cdot X^5$

**Notice that now we have a sum of powers with the same base, in this case, we do not add the exponents, we simply simplify the like terms, that is, we simply subtract to obtain:**

$4X^5-2X^5=$

$2X^5$

Do you know what the answer is?

### Let's look at another example

$3\cdot X^4\cdot4\cdot X^2\cdot X=$

Pay attention, in this exercise there is a multiplication among all the terms.

We will proceed according to the properties we learned: if we have the same base $X$ with a multiplication operation between each base, we can add the exponents. When there is no exponent it means that the base is raised to the power $1$.

**We will do it and obtain:**

$3\cdot X^7\cdot4=$

**Excellent. Now, we can multiply** **$3$**** by** **$4$**** and obtain:**

$12\cdot X^7=$

**Undoubtedly we can multiply the** **$X$**** by its coefficient and obtain:**

$12X^7=$

**One last exercise where you must solve for the variable** **$X$**

$4^4\cdot4^2\cdot4^x=4^9$

Without using a calculator, we can work according to the technique we have learned, adding the exponents of the same base among multiplication and equalizing the $X$ in the exponent to the exponent on the right side.

**We will start by adding the exponents and obtain:**

$4^{6+x}=4^9$

For the equation to be correct the exponents must be equal since it is the same base. Therefore, we will compare the exponents and solve for $X$. We will obtain:

$6+X=9$

$X=3$

**Important:**

Not only does the law of exponents for products with the same base exist, there is also a law for division of powers with the same base (quotient of powers with the same base). Properly handling it will allow us to simplify algebraic expressions and solve different types of equations.

But remember that the product and quotient law only apply when the operation involves the same bases, and not when we have multiplication of powers with different bases or division of powers with different bases.

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

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

## Multiplication Exercises of Powers with the Same Base

### Exercise 1

**Solve the following exercise:**

$4^2\times4^4=$

**Solution**

According to the power property, when there are two powers with the same base they are multiplied by each other. It is necessary to add the power coefficient.

$2+4=6$

**Answer:**

Therefore, the solution is:

$4^6$

### Exercise 2

**Solve the following exercise:**

$5^4\times25=$

**Solution**

In this exercise, we must first identify that the number $25$ can be broken down into its power form, which is $5^2$.

Once we did this, we can operate again according to the power rule and solve: $4+2=6$

**Answer:**

The solution: $5^6$

### Exercise 3

**Solve the following exercise:**

$7^9\times7=$

**Solution**

According to the power property, when there are two powers with the same base they are multiplied by each other. It is necessary to add the power's coefficient.

It is important to remember that a number without a power has a value equal to the power of $1$, and not $0$.

Therefore: $9+1=10$

**Answer:**

The solution: $7^{10}$

Do you think you will be able to solve it?

### Exercise 4

**Solve the following exercise:**

$2^{10}\times2^7\times2^6=$

**Solution**

Also, when there are a number of products, even when multiplied by each other, the operation between the power coefficients will be the sum.

$10+7+6=23$

**Answer:**

Therefore, the solution is:

$2^{23}$

### Exercise 5

**Homework:**

Simplify the expression:

$a^3\times a^2\times b^4\times b^5=$

**Solution**

It's important to remember that according to the power rule for multiplication, you can only add the exponents when they have the same base. Therefore, we add the exponents of a separately from those of $b$.

Therefore

$3+2=5$

$4+5=9$

$a^5\times b^9$

**Answer:**

$a^5\times b^9$

## Review Questions

**How to multiply powers with the same base?**

When we perform a multiplication of powers, and these have the same bases, we must add the exponents, the sum will be the new exponent and the base remains the same.

**How to solve multiplications of powers with different bases?**

In this case, the exponents cannot be added.

**What happens when a base does not have an exponent?**

When a number or expression does not have an exponent, it is said to have an exponent $1$.

**What does it mean for a base to be raised to the** **$0$****?**

If a number or expression different from zero is raised to zero, the result is $1$.

Do you know what the answer is?

## Examples with solutions for Multiplication of Powers

### Exercise #1

### Video Solution

### Step-by-Step Solution

To solve the exercise we use the property of multiplication of powers with the same bases:

$a^n * a^m = a^{n+m}$

With the help of this property, we can add the exponents.

$4^2\times4^4=4^{4+2}=4^6$

### Answer

### Exercise #2

### Video Solution

### Step-by-Step Solution

According to the property of powers, when there are two powers with the same base multiplied together, the exponents should be added.

According to the formula:$a^n\times a^m=a^{n+m}$

It is important to remember that a number without a power is equivalent to a number raised to 1, not to 0.

Therefore, if we add the exponents:

$7^{9+1}=7^{10}$

### Answer

### Exercise #3

$2^{10}\cdot2^7\cdot2^6=$

### Video Solution

### Step-by-Step Solution

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

Keep in mind that all the terms in the multiplication have the same base, so we will use the previous property:

$2^{10}\cdot2^7\cdot2^6=2^{10+7+6}=2^{23}$**Therefore, the correct answer is option c.**

### Answer

### Exercise #4

### Video Solution

### Step-by-Step Solution

All bases are equal and therefore the exponents can be added together.

$8^2\cdot8^3\cdot8^5=8^{10}$

### Answer

### Exercise #5

### Video Solution

### Step-by-Step Solution

To solve this exercise, first we note that 25 is the result of a power and we reduce it to a common base of 5.

$\sqrt{25}=5$$25=5^2$Now, we go back to the initial exercise and solve by adding the powers according to the formula:

$a^n\times a^m=a^{n+m}$

$5^4\times25=5^4\times5^2=5^{4+2}=5^6$

### Answer