How to Add Exponents

Adding Numbers With Exponents By Hand

Solve the first exponential expression. An exponential expression has a base (large number) and exponent (small number). The exponent tells you how many times to multiply the base by itself ( 2 3 = 2 × 2 × 2 {\displaystyle 2^{3}=2\times 2\times 2} 2^{{3}}=2\times 2\times 2). For example, if your problem is 3 4 + 2 5 {\displaystyle 3^{4}+2^{5}} 3^{{4}}+2^{{5}}, you would first calculate 3 4 {\displaystyle 3^{4}} 3^{{4}}: 3 4 {\displaystyle 3^{4}} 3^{{4}} = 3 × 3 × 3 × 3 {\displaystyle =3\times 3\times 3\times 3} =3\times 3\times 3\times 3 = 81 {\displaystyle =81} =81

Solve the second exponential expression. To do this, multiply the base by itself the number of times indicated by the exponent. For example, the problem is now 81 + 2 5 {\displaystyle 81+2^{5}} 81+2^{{5}}, so you need to calculate 2 5 {\displaystyle 2^{5}} 2^{{5}}: 2 5 {\displaystyle 2^{5}} 2^{{5}} = 2 × 2 × 2 × 2 × 2 {\displaystyle =2\times 2\times 2\times 2\times 2} =2\times 2\times 2\times 2\times 2 = 32 {\displaystyle =32} =32

Add the two values together. This will give you the sum of the two exponential expressions. For example: 3 4 + 2 5 {\displaystyle 3^{4}+2^{5}} 3^{{4}}+2^{{5}} = ( 3 × 3 × 3 × 3 ) + ( 2 × 2 × 2 × 2 × 2 ) {\displaystyle =(3\times 3\times 3\times 3)+(2\times 2\times 2\times 2\times 2)} =(3\times 3\times 3\times 3)+(2\times 2\times 2\times 2\times 2) = ( 81 ) + ( 32 ) {\displaystyle =(81)+(32)} =(81)+(32) = 113 {\displaystyle =113} =113

Adding Numbers With Exponents Using a Calculator

Locate the exponent key on your calculator. This key will likely look like y x {\displaystyle y^{x}} y^{{x}} or E X P {\displaystyle EXP} EXP, or it may look like an x {\displaystyle x} x with a blank box as the exponent. If you do not have a scientific calculator, you cannot use this method.

Type in the first exponential expression. To do this, hit the base number (large number) first, then hit the exponent. For example, if your problem is 3 4 + 2 5 {\displaystyle 3^{4}+2^{5}} 3^{{4}}+2^{{5}}, you would hit the following sequence of keys to solve the first expression: 3 {\displaystyle 3} 3 y x {\displaystyle y^{x}} y^{{x}} 4 {\displaystyle 4} 4

Hit the addition key. This will show you the value of the first exponential expression. You do not need to hit the equal key ( = {\displaystyle =} =) after typing in the first exponential expression. For example, after typing in the expression 3 4 {\displaystyle 3^{4}} 3^{{4}}, you should hit the + {\displaystyle +} + symbol to see a value of 81 {\displaystyle 81} 81.

Type in the second exponential expression. To do this, hit the base number (large number) first, then hit the exponent. For example, if your problem is 3 4 + 2 5 {\displaystyle 3^{4}+2^{5}} 3^{{4}}+2^{{5}}, you would hit the following sequence of keys to solve the second expression: 2 {\displaystyle 2} 2 y x {\displaystyle y^{x}} y^{{x}} 5 {\displaystyle 5} 5

Hit the equal key ( = {\displaystyle =} =). This will show you the final sum of the two exponential expressions. For example, after hitting the appropriate sequence of keys, 3 4 + 2 5 {\displaystyle 3^{4}+2^{5}} 3^{{4}}+2^{{5}} adds up to 113 {\displaystyle 113} 113.

Adding Variables With Exponents

Find terms with the same base and the same exponent. The base is the large number (or variable) in the exponential expression, and the exponent is the small number. The exponent tells you how many times to multiply the base by itself ( x 3 = x × x × x {\displaystyle x^{3}=x\times x\times x} x^{{3}}=x\times x\times x). In the case of variables, an exponential expression will also have a coefficient, which is a number appearing before the variable that tells you how to multiply the variable. Even if a variable has no coefficient, it is understood to have the coefficient of 1 {\displaystyle 1} 1. For example, x 4 = 1 x 4 {\displaystyle x^{4}=1x^{4}} x^{{4}}=1x^{{4}}

Add the terms with the same base and exponent. When working with variables, there is no way to add terms that do not have the same base and the same exponent. The terms must have BOTH of these parts in common. For example, if the problem is x 4 + 3 x 6 + 4 x 4 + 2 y 4 {\displaystyle x^{4}+3x^{6}+4x^{4}+2y^{4}} x^{{4}}+3x^{{6}}+4x^{{4}}+2y^{{4}}, you should note that x 4 {\displaystyle x^{4}} x^{{4}} and 4 x 4 {\displaystyle 4x^{4}} 4x^{{4}} have the same base ( x {\displaystyle x} x) and the same exponent ( 4 {\displaystyle 4} 4). Thus, these two terms can be added together. The term 3 x 6 {\displaystyle 3x^{6}} 3x^{{6}} has a different exponent, so it cannot be added; the term 2 y 4 {\displaystyle 2y^{4}} 2y^{{4}} has a different base, so it cannot be added.

Add the coefficients of the like terms. Remember, if a term has no coefficient shown, a coefficient of 1 {\displaystyle 1} 1 is understood. Do NOT add the exponents. The exponent stays the same. For example, if you are calculating x 4 + 4 x 4 {\displaystyle x^{4}+4x^{4}} x^{{4}}+4x^{{4}} you would add together the coefficients, and x 4 {\displaystyle x^{4}} x^{{4}} would stay the same: x 4 + 4 x 4 {\displaystyle x^{4}+4x^{4}} x^{{4}}+4x^{{4}} = ( 1 ) x 4 + ( 4 ) x 4 {\displaystyle =(1)x^{4}+(4)x^{4}} =(1)x^{{4}}+(4)x^{{4}} = 5 x 4 {\displaystyle =5x^{4}} =5x^{{4}}

Write out the final, simplified addition sentence. Remember, you cannot add exponential expressions that do not have the same base AND exponent, so those will stay the same as they were in the original problem. For example, x 4 + 3 x 6 + 4 x 4 + 2 y 4 {\displaystyle x^{4}+3x^{6}+4x^{4}+2y^{4}} x^{{4}}+3x^{{6}}+4x^{{4}}+2y^{{4}} simplifies to 5 x 4 + 3 x 6 + 2 y 4 {\displaystyle 5x^{4}+3x^{6}+2y^{4}} 5x^{{4}}+3x^{{6}}+2y^{{4}}.