Table of Contents

Last modified on October 26th, 2024

Exponents cannot be added or subtracted directly. Only coefficients or variables with the same base and power can be added or subtracted.

The conditions for adding or subtracting exponent terms are:

- All expressions must have the same base
- All expressions must have equal exponents

If both statements are true, the expressions can be added or subtracted by combining like terms.

There are two possible cases:

When adding the exponents with the same base, the coefficients of each expression are added together by keeping the base and the exponent the same.

Let us add 6x^{5} + 4x^{5}

Here,

The base is x, which is the same.

The exponent is 5, which is the same.

Now, on adding the coefficients, we get

6 + 4 = 10

Thus, the sum is

6x^{5} + 4x^{5} = (6 + 4)x^{5} = 10x^{5}

**Add the following expression:****2 × 7 ^{3} + 5 × 7^{3} + 9 × 7^{3}**

Solution:

Here, 2 × 7^{3} + 5 × 7^{3} + 9 × 7^{3}

= (2 + 5 + 9) × 7^{3}

= 16 × 7^{3}

Thus, the sum is 16 × 7^{3}

**Problem: **Adding expressions with **VARIABLES**

**Add: a ^{6} + 4a^{6} + 3a^{6}**

Solution:

Here, a^{6} + 4a^{6} + 3a^{6}

= (1 + 4 + 3)a^{6}

= 8a^{6}

Thus, the sum is 8a^{6}

However, when adding the exponents with different bases and exponents, we first calculate each exponent and then add them to get the sum.

For example, 2^{3} + 5^{2} = (2 × 2 × 2) + (5 × 5) = 8 + 25 = 33

**Add 7 ^{2} + 3^{4}**

Solution:

Here, 7^{2} + 3^{4}

= (7 × 7) + (3 × 3 × 3 × 3)

= 49 + 81

= 130

Thus, the sum is 130

Adding negative exponents is done by calculating each term separately and then adding the total. The general form of adding negative exponents with different bases is:

**x**^{-m}** + y**^{-n}** = ${\dfrac{1}{x^{m}}+\dfrac{1}{y^{n}}}$ **

For example,

2^{-1} + 3^{-2} = ${\dfrac{1}{2^{1}}+\dfrac{1}{3^{2}}}$ = ${\dfrac{1}{2}+\dfrac{1}{9}}$ = ${\dfrac{11}{18}}$

**Add the following expressions:****a) 2 ^{-3} + 2^{-2}**

Solution:

a) Here, 2^{-3} + 2^{-2}

= ${\dfrac{1}{2^{3}}+\dfrac{1}{2^{2}}}$

= ${\dfrac{1}{8}+\dfrac{1}{4}}$ (since 2^{3} = 8 and 2^{2} = 4)

= ${\dfrac{1+2}{8}}$

= ${\dfrac{3}{8}}$

Thus, the sum is ${\dfrac{3}{8}}$

b) Here, 5^{-2} + 4^{-1}

= ${\dfrac{1}{5^{2}}+\dfrac{1}{4^{1}}}$

= ${\dfrac{1}{25}+\dfrac{1}{4}}$ (since 5^{2} = 25 and 4^{1} = 4)

= ${\dfrac{4+25}{100}}$

= ${\dfrac{29}{100}}$

Thus, the sum is ${\dfrac{29}{100}}$

Fractional exponents are added by calculating each term separately and then adding the total. The general form of adding fractional exponents is expressed as: **${x^{\dfrac{n}{m}}+y^{\dfrac{p}{q}}}$**

For example,

${64^{\dfrac{1}{2}}+125^{\dfrac{1}{3}}}$

= ${\sqrt{64}+\sqrt[3] {125}}$

= 8 + 5 = 13

**Solve: ${3a^{\dfrac{3}{4}}+\left( -2\right) a^{\dfrac{3}{4}}}$**

Solution:

Here, ${3a^{\dfrac{3}{4}}+\left( -2\right) a^{\dfrac{3}{4}}}$

= ${3a^{\dfrac{3}{4}}-2a^{\dfrac{3}{4}}}$

= ${\left( 3-2\right) a^{\dfrac{3}{4}}}$

= ${a^{\dfrac{3}{4}}}$

Thus, the sum is ${a^{\dfrac{3}{4}}}$

The exponent rules for subtraction are the same as the exponent rules for addition. In order to subtract the terms with exponents, the bases and the exponents of all terms must be the same. It can also be interpreted as adding negative terms with exponents.

When subtracting the exponents with the same base, the coefficients of each expression are subtracted together by keeping the base and the exponent the same.

**Simplify: 9 ^{4} – 9^{2}**

Solution:

Here, 9^{4} – 9^{2}

= (9 × 9 × 9 × 9) – (9 × 9)

= 6561 – 81

= 6480

Thus, the difference is 6480

**Problem**:** **Subtracting expressions with **VARIABLES**

**Subtract: 5x ^{6} – 8x^{6}**

Solution:

Here, 5x^{6} – 8x^{6}

= (5 – 8)x^{6}

= -3x^{6}

Thus, the difference is -3x^{6}

However, when subtracting the exponents with different bases and exponents, we first calculate each exponent and then subtract them to get the difference.

For example, 2^{3} – 5^{2} = (2 × 2 × 2) – (5 × 5) = 8 – 25 = -17

**Subtract: 7 ^{2} – 3^{4}**

Solution:

Here, 7^{2} – 3^{4}

= (7 × 7) – (3 × 3 × 3 × 3)

= 49 – 81

= -32

Thus, the difference is -32

Last modified on October 26th, 2024