# Adding and Subtracting Exponents

In mathematics, exponents are used to representing a larger number in terms of power. It indicates how many times the base is multiplied by itself. Where a base is any number or any mathematical expression. For example, A^{3} here base is A and power is 3 which means A will multiply by itself three times that is A^{3} = A x A x A. The general term of the exponent is

Y

^{n}= Y × Y × Y ×………n times

Here, y is known as base, and n is known as power or exponent. Types of exponents:

**Negative Exponent:**Negative exponents are those exponents which tell how many times the reciprocal of the base multiples with itself. It is represented like a^{-n}or 1/a^{n}. For example, 23^{-2}, 4^{-2}.**Fractional Exponent:**When an exponent is represented in terms of fraction then such types of exponents are known as fractional exponents. It is represented like a^{1/n}. For example, 3^{1/2}, 4^{1/3}.**Decimal Exponent:**When an exponent is represented in terms of decimal digits then such types of exponents are known as decimal exponents. It is represented like a^{1.3}. For example, 3^{1.5}, 4^{12.3}.

### Adding and Subtracting Exponents

Addition and subtraction are the two basic operations of mathematics. The addition means finding the sum of two digits and subtraction means finding the difference between two digits. But we cannot directly add or subtract exponents, we can only perform addition or subtraction only on the coefficients or variables that have the same base and the same power. We can only add exponents in multiplication and subtract exponents in the division.

**Steps to do addition or subtraction between exponents in algebra:**

To do addition or subtraction we will follow the same steps:

Step 1:Before performing any addition or subtraction between exponents we need to observe whether the base and exponents are the same or not.

Step 2:Arrange the similar variables/terms together.

Step 3:Now perform addition or subtraction as per need between the coefficient of terms.

**Example 1: Solve 6x ^{3 }+ 12x^{3}.**

**Solution:**

Here the base is same i.e., x and exponents of two terms are also same i.e., 3

So we can add coefficients of the two terms to get result.

6x

^{3 }+ 12x^{3 }= (6 + 12)x^{3}= 18x

^{3}

**Example 2: Solve 9x ^{3 }-13x^{3}.**

**Solution:**

Here the base is same i.e., x and exponents of two terms are also same i.e., 3

So we can subtract coefficients of the two terms to get result.

9x

^{3 }– 13x^{3 }= (9 – 13)x^{3}= -4x

^{3}

**Steps to do addition or subtraction between exponents in numbers:**

To do addition or subtraction we will follow the same steps:

Step 1:Before performing any addition or subtraction between exponents we need to observe whether the base and exponents are the same or not.

Step 2:Arrange the similar base and exponent terms together. If the terms have different base and exponent then solve them individually.

Step 3:Now perform addition or subtraction as per need between the base of terms.

**Example 1: Solve 6 ^{3 }+ 6^{3}.**

**Solution:**

Here the base is same i.e., 6 and exponents of two terms are also same i.e., 3

So, we are solving them together

6

^{3 }+ 6^{3 }= 2(6)^{3}= 2 x 6 x 6 x 6

= 432

**Example 2: Solve 9 ^{2 }– 13^{3}.**

**Solution:**

Here the base and the exponents are different

So, we are solving them individually.

9

^{2 }– 13^{3 }= 9 x 9 -13 x 13 x 13= 81 – 2197

= -2116

### Similar Questions

**Question 1: Solve 5x ^{3 }+ 3x^{3}.**

**Solution:**

Here the base is same i.e., x and exponents of two terms are also same i.e., 3

So we can add coefficients of the two terms to get result.

5x

^{3 }+ 3x^{3 }= (5 + 3)x^{3}= 8x

^{3}So, 5x

^{3 }+ 3x^{3 }= 8x^{3}

**Question 2: What is the result of expression -11a ^{2 }+ 4a^{2}.**

**Solution:**

Here the base is same i.e., a and exponents of two terms are also same i.e., 2

So we can add coefficients of the two terms to get result.

-11a

^{2 }+ 4a^{2 }= (-11 + 4)a^{2}= (4 – 11)a

^{2}= -7a

^{2}So, -11a

^{2 }+ 4a^{2 }= -7a^{2}

**Question 3: Solve the expression 4x ^{3 }+ 4x^{2 }– 2x^{3 }+ x^{2 }– x + 1**

**Solution:**

Here we have different kind of terms (x

^{3}, x^{2}, x)i.e., bases are same but different exponents.So identify the similar kind of variables and group them and perform addition/subtraction based on signs and arrange in polynomial order i.e., bases having higher exponential at first and lower at last.

4x

^{3 }+ 4x^{2 }– 2x^{3 }+ x^{2 }– x + 1 = (4x^{3 }– 2x^{3}) + (4x^{2 }+ x^{2})-x + 1= (4 – 2)x

^{3 }+ (4 + 1)x^{2 }– x + 1= 2x

^{3 }+ 5x^{2 }– x + 1

**Question 4: What is the result of x ^{3}y + 4x^{3}y?**

**Solution:**

Here we have 2 different variables in 2 terms x, y and the exponents of x, y in two terms are same i.e., 3,1 respectively.

So we can consider that this 2 terms has matching variables and we can add/subtract the coefficient of 2 terms based on requirement.

x

^{3}y + 4x^{3}y = (1 + 4)x^{3}y= 5x

^{3}y

**Question 5: Solve x ^{3}y + 4x^{3}y^{2 }+ 4x – x + 1**

**Solution:**

Here we have 2 different variables in 2 terms x, y and the exponent of x in 2 terms are same i.e., 3 but the exponent of y is not same so we can’t consider the 2 terms as same and we won’t perform any operation between them. (remained as it is)

But there are other two more terms with same base variable x and exponent as 1. So we group them and perform computations on its coefficients.

x

^{3}y + 4x^{3}y^{2 }+ 4x – x + 1 = 4x^{3}y^{2 }+ x^{3}y + (4x – x) + 1= 4x

^{3}y^{2 }+ x^{3}y + (4 – 1)x + 1= 4x

^{3}y^{2 }+ x^{3}y + 3x + 1

**Question 6: Solve x ^{5}y^{2 }– x^{4}y^{4}**

**Solution:**

Here there are two terms x

^{5}y^{2}, x^{4}y^{4}and in two terms we have 2 variables x, y but the exponents of variables are not same when made comparison between terms.Hence the first term is entirely not like the second term and can’t be subtracted from each other.

We leave them as it is.

x

^{5}y^{2 }– x^{4}y^{4}can’t be simplified further.