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Adding powers with the same base and different exponents

How do you add powers with the same base and different exponents?

See the following examples.

Examples of adding powers with the same base and different exponents

1. Adding two powers with the same base and different exponents

35 + 36 =
35 + 35 + 1 =
35 + 35 * 31 =
35(1 + 31) =
4 * 35 =
4 * 243 = 972

See Powers of 3.

2. Adding exponential expressions with the same base and different exponents

45 + 48 =
45 + 45 + 3 =
45 + 45 * 43 =
45(1 + 43) =
1024(1 + 64) =
1024 * 65 = 66560

See Powers of 4.

3. Adding powers with the same negative base and unequal exponents

(-2)3 + (-2)9 =
(-2)3 + (-2)3 + 3 + 3 =
(-2)3 + (-2)3 * (-2)3 * (-2)3 =
(-2)3(1 + (-2)3 * (-2)3) =
-8(1 + (-8) * (-8)) =
-8(1 + 64) = -520

See Powers of 2.

4. Addition of powers with the same positive base and unequal exponents

ab + a5b =
ab + a5 * b =
ab + a5 * ab =
ab(1 + a5)