Exponents cannot be added or subtracted directly. Only coefficients or variables with the same base and power can be added or subtracted.
Rules
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.
Adding Exponents
There are two possible cases:
With the Same Base
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 6x5 + 4x5
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
6x5 + 4x5 = (6 + 4)x5 = 10x5
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Add the following expression:
2 × 73 + 5 × 73 + 9 × 73
Solution:
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Here, 2 × 73 + 5 × 73 + 9 × 73
= (2 + 5 + 9) × 73
= 16 × 73
Thus, the sum is 16 × 73
Problem: Adding expressions with VARIABLES
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Add: a6 + 4a6 + 3a6
Solution:
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Here, a6 + 4a6 + 3a6
= (1 + 4 + 3)a6
= 8a6
Thus, the sum is 8a6
With Different Bases and Powers
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, 23 + 52 = (2 × 2 × 2) + (5 × 5) = 8 + 25 = 33
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Add 72 + 34
Solution:
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Here, 72 + 34
= (7 × 7) + (3 × 3 × 3 × 3)
= 49 + 81
= 130
Thus, the sum is 130
Negative Exponents
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 = ${\dfrac{1}{2^{1}}+\dfrac{1}{3^{2}}}$ = ${\dfrac{1}{2}+\dfrac{1}{9}}$ = ${\dfrac{11}{18}}$
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Add the following expressions:
a) 2-3 + 2-2
b) 5-2 + 4-1
Solution:
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a) Here, 2-3 + 2-2
= ${\dfrac{1}{2^{3}}+\dfrac{1}{2^{2}}}$
= ${\dfrac{1}{8}+\dfrac{1}{4}}$ (since 23 = 8 and 22 = 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 52 = 25 and 41 = 4)
= ${\dfrac{4+25}{100}}$
= ${\dfrac{29}{100}}$
Thus, the sum is ${\dfrac{29}{100}}$
Fractional Exponents
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
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Solve: ${3a^{\dfrac{3}{4}}+\left( -2\right) a^{\dfrac{3}{4}}}$
Solution:
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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 ${a^{\dfrac{3}{4}}}$
Subtracting Exponents
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.
With the Same Base
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.
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Simplify: 94 – 92
Solution:
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Here, 94 – 92
= (9 × 9 × 9 × 9) – (9 × 9)
= 6561 – 81
= 6480
Thus, the difference is 6480
Problem: Subtracting expressions with VARIABLES
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Subtract: 5x6 – 8x6
Solution:
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Here, 5x6 – 8x6
= (5 – 8)x6
= -3x6
Thus, the difference is -3x6
With Different Bases and Powers
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, 23 – 52 = (2 × 2 × 2) – (5 × 5) = 8 – 25 = -17
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Subtract: 72 – 34
Solution:
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Here, 72 – 34
= (7 × 7) – (3 × 3 × 3 × 3)
= 49 – 81
= -32
Thus, the difference is -32
