Adding Exponents – Visual Fractions

Adding exponents is a crucial skill to learn in your algebra class. Knowing how to add exponents is essential in more advanced math topics and classes. This is why it’s important that you master this topic or at least, take a refresher before tackling more complex topics.

The base and exponent of a term are the two most important components that you need to check when adding exponents or terms with exponents. In this article, you’ll learn the reasoning behind this statement. You’ll also know how to add terms and the crucial steps needed when adding exponents. Get your notebook or scratchpad ready, we’ve prepared a lot of problems for you to work on as well!

What Is Adding Exponents?

Adding exponents is the process of combining terms with exponents given that they share the same base and exponent. When adding exponents, it is important to identify the components correctly. Recall that when given a term with exponents, $bx^n$ , b is the coefficient, x is the term, and n is the exponent. For numbers expressed as powers, it only contains a base and an exponent.

Recall that exponents represent the power that a given number or base is raised to. For $4^6$ , it’s equivalent to 4 being multiplied 6 times in a row. It’s much easier to express this through exponents, thus, the importance of knowing when it’s possible to add two exponents.

\begin{aligned}{\color{Purple} 4}^{\color{DarkBlue} 6} &= \underbrace{\color{Purple} 4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4}_{\color{DarkBlue} 6}\\5{\color{DarkRed}x}^{\color{DarkOrange} 3} &= 5\,\cdot\underbrace{\color{DarkRed} x\cdot x\cdot x}_{\color{DarkOrange} 3}\end{aligned}

This reasoning applies to algebraic terms with exponents as well. The expression, $5x^3$ , the term, $x^3$ , tells us that x is multiplied three times in a row. Now, adding exponents is simply combining terms with exponents if they meet the set condition: they share the same base and exponent. Check out the table below to help you understand which terms can be added together.

Terms That Can Be Added	Terms That Can’t Be Added
$3^2 + 3^2 = 2(3^2)$	$3^2 + 2^3$
$(-4)^3 + 3(-4)^3 = 4(-4)^2$	$(-6)^2 + (-5)^2$
$2x^6 + 3x^6 = 5x^6$	$2x^3 + 2y^3$
$-5y^8 + 6y^8 =y^8$	$-4x^4 + 5y^5$

This shows that when adding exponents, the first crucial step that you need to make is to check whether the terms can actually be added. Before heading over to the next section, test your understanding first by trying out the few examples shown below.

Problem 1

Determine whether each pair of terms with exponents can be added together.

a. $2(7^5)$ and $3(7^5)$

When checking whether two terms can be added, inspect the base and exponent. Double-check whether the bases and exponents are equal. If they are, the two terms can be added.

$2({\color{Teal}7}^{\color{Orchid}5})$ : It has a base of 7 and a power of 5.
$3({\color{Teal}7}^{\color{Orchid}5})$ : It has a base of 7 and a power of 5.

This confirms that the method of adding exponents can be applied on these two terms. In fact, the sum of these two terms is equal to $2(7^5) + 3(7^5) = 5(7^5)$ .

b. $2(5^3)$ and $5(2^3)$

Apply a similar process to verify whether the two terms can be added.

$2({\color{DarkGreen}5}^{\color{DarkOrange}3})$ : It has a base of 5 and an exponent of 3.
$5({\color{DarkGreen}2}^{\color{DarkOrange}3})$ : It has a base of 2 and an exponent of 3.

Although the two terms share the same exponents, their bases are not equal. This means that the two terms with exponents can’t be added together.

c. $-x^4$ and $4x^3$

This time, let’s take a look at exponents with algebraic terms. Keep in mind that for cases like this, the base would be the algebraic term, x. Inspect the rest of the components to see whether the two terms can be added.

$-1({\color{Purple}x}^{\color{Magenta}4})$ : It has a base of x and an exponent of 4.
$4({\color{Purple}x}^{\color{Magenta}3})$ : It has a base of x and an exponent of 3.

Since the exponents are not the same, the two terms do not meet the conditions needed when adding exponents. This means that the algebraic terms, $-x^4$ and $4x^3$ , can’t be combined together even more.

Now that you’re confident in assessing terms, it’s time to learn how to add exponents. Use what you’ve learned in the earlier section and add the terms that share the same base and exponent.

What Are the Steps When Adding Exponents?

There are two crucial steps when adding exponents: 1) first is making sure that the two terms with exponents can be added and 2) applying the right technique depending on the answer from the first step.

When adding terms with the same base and exponent, add the coefficients and simplify the sum as one term.
When adding terms not sharing the same base or exponent, the only way to find their sum is by evaluating each term first and then adding the two values.

The examples below cover different conditions, so test your understanding by going through the different problems. Use our discussion to work on the items independently while using our discussion as a guide when you need to. Let’s begin with the simplest example – adding two terms that meet our desired conditions.

Problem 2

What is the sum of $\boldsymbol{2(2^3)}$ and $\boldsymbol{3(2^3)}$ ?

For the first case, let’s see what happens when we add terms with the same exponent and base. By inspection, you can see that the two terms share the same base and exponent.

$2({\color{Teal}2}^{\color{Orchid}3})$ : It has a base of 2 and a power of 3.
$3({\color{Teal}2}^{\color{Orchid}3})$ : It has a base of 2 and a power of 3.

When working with terms like these, we treat the term with exponents like algebraic terms. This means that when adding the terms containing the exponents; simply combine the two into one term by adding the coefficients before each term.

\begin{aligned}{\color{Green}2}(2^3)+ {\color{Green}3}(2^3) &= {\color{Green}(2 +3)}(2^3)\\&= 5(2^3)\end{aligned}

This means that the sum of $2(2^3)$ and $3(2^3)$ is equal to $5(2^3)$ .

Problem 3

What is the sum of $\boldsymbol{5(x^4)}$ and $\boldsymbol{6(x^4)}$ ?

When working with algebraic terms sharing the same exponent and base in its variables, apply a similar process. Of course, don’t forget to double-check the terms to see if they satisfy the given condition.

$5({\color{DarkGreen}x}^{\color{DarkOrange}4})$ : It has a base of x and an exponent of 4.
$6({\color{DarkGreen}x}^{\color{DarkOrange}4})$ : It has a base of x and an exponent of 4.

This means that for you to find the sum of the two given terms, focus on adding the coefficients of the terms.

\begin{aligned}{\color{Green}5}(x^4)+ {\color{Green}6}(x^4) &= {\color{Green}(5 +6)}(x^4)\\&= 11(x^4)\\&= 11x^4\end{aligned}

Hence, the sum of $5x^4$ and $6x^4$ is equal to $11x^4$ .

Problem 4

What is the sum of $\boldsymbol{3(2^2)}$ and $\boldsymbol{4(3^2)}$ ?

Now, let’s address the other methods of adding terms with exponents. This problem addresses our question: “What if the terms do not share the same terms and bases? Can’t these terms be added?” The quick answer to that is yes, but, you’ll need to apply a different approach.

$3({\color{Purple}2}^{\color{Magenta}2})$ : It has a base of 2 and an exponent of 2.
$4({\color{Purple}3}^{\color{Magenta}2})$ : It has a base of 3 and an exponent of 2.

This confirms that the two terms don’t share the same bases, so they can’t be added by adding the terms’ coefficients. Instead, what you’ll do is evaluate the value of each term and then add the resulting values together.

\begin{aligned}3(2^2) + 4(3^2) &= 3(2 \cdot 2) + 4(3\cdot 3\cdot 3)\\&= 3(4) + 4(9)\\&= 12 + 36\\&= 48\end{aligned}

This shows that even when the terms do share the same base, exponent, or both, there is still a way to add numbers with exponents. In fact, the sum of $3(2^2)$ and $4(3^2)$ is equal to 48.

Problem 5

What is the sum of $\boldsymbol{5y^4}$ and $\boldsymbol{5y^3}$ ?

Will the method from Problem 4 apply when you’re working with algebraic terms? Unfortunately, the rule for adding exponents is strictly applied to algebraic terms. This means that algebraic terms with exponents can only be added together if and only if they share the same base and exponent.

$5({\color{Teal}y}^{\color{Orchid}4})$ : It has a base of y and a power of 4.
$5({\color{Teal}y}^{\color{Orchid}3})$ : It has a base of y and a power of 3.

When you’re working with algebraic terms that don’t share the same base and exponent, simply express the sum as a binomial. Feel free to factor out the common factor shared by the terms’ coefficients.

\begin{aligned}5y^4 + 5y^3 &= 5y^4 + 5y^3\\&= 5(y^4 +y^3)\end{aligned}

Hence, the sum of $5y^4$ and $5y^3$ is equal to $5(y^4 +y^3)$ .

We’ve now covered all the possible bases for you, so it’s time for you try more complex examples when it comes to adding exponents.

Problem 6

What is the sum of $\boldsymbol{3^{-2}}$ and $\boldsymbol{2^{-3}}$ ?

When adding terms with exponents, whether positive or negative, the first step is always inspecting whether the base and exponent are identical.

${\color{DarkGreen}3}^{\color{DarkOrange}-2}$ : It has a base of 3 and an exponent of -2.
${\color{DarkGreen}2}^{\color{DarkOrange}-3}$ : It has a base of 2 and an exponent of -3.

Since the two terms don’t share the same base and exponent, the best approach is to evaluate each term and add the resulting values. When working with negative exponents, remember that $b^{-m} = \dfrac{1}{b^m}$ .

\begin{aligned}3^{-2} + 2^{-3}&= \dfrac{1}{3^2}+ \dfrac{1}{2^3}\\&= \dfrac{1}{9} + \dfrac{1}{8}\\&= \dfrac{17}{72}\end{aligned}

This means that $3^{-2}$ and $2^{-3}$ is equal to $\frac{17}{72}$ .

Problem 7

What is the evaluated form of $2x^4 + 5y^3 + 3x^4 + 2y^{-3}$ ?

When given a group of terms, identify the terms that share the same base. Apply appropriate rules when adding exponents and only combine terms when they share the same base and exponent.

\begin{aligned}2x^4 + 5y^3 + 3x^4 + 2y^{-3}&= (2x^4 + 3x^4) + (5y^3 + 2y^{-3})\\&=[({\color{Green}2 + 3})x^4] + (5y^3 +2y^{-3})\\&= 5x^4 + (5y^3+ 2y^{-3})\end{aligned}

Now, to simplify the sum of the two terms with y, rewrite $y^{-3}$ as a fraction with a positive exponent.

\begin{aligned}5x^4 + (5y^3+ 2y^{-3})&= 5x^4 + 5y^3 + \dfrac{2}{y^3}\end{aligned}

This answer is already correct, especially if you haven’t learned how to add rational expressions yet. But if you already know how to add rational expressions, you can still further simplify the sum of the expression.

\begin{aligned}5x^4 + 5y^3 + \dfrac{2}{y^3} &= \dfrac{5x^4(y^3) +5y^3(y^3) + 2}{y^3}\\&= \dfrac{5x^4y^3 +5y^6 + 2}{y^3}\\&= \dfrac{1}{y^3}(5x^4y^3 + 5y^6 + 2)\end{aligned}

This means that $2x^4 + 5y^3 + 3x^4 + 2y^{-3}$ is equivalent to $5x^4 + 5y^3 + \dfrac{2}{y^3}$ or $\dfrac{1}{y^3}(5x^4y^3 + 5y^6 + 2)$ .