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Adding powers with different bases and negative exponents

How to add powers with different bases and negative exponents?

See the following examples.

Examples of adding powers with different bases and negative exponents

1. Add different base and negative exponents

2-6 + 4-6 =
2-6 + (2 * 2)-6 =
2-6 + (2-6 * 2-6) =
2-6(1 + 2-6) =
(1/26) * (1 + 1/26) =
(1/64) * (1 + 1/64) =
1/64 * 65/64 = 65/4096

See Powers of 2.

2. Adding two powers with different bases and negative exponents

3-2 + 12-2 =
3-2 + (3 * 4)-2 =
3-2 + (3-2 * 4-2) =
3-2(1 + 4-2) =
(1/32) * (1 + 1/42) =
(1/9) * (1 + 1/16) =
(1/9) * (17/16) = 17/144

See Powers of 4.

3. Addition of exponential expressions with different bases and negative exponents

(ab)-c + a-c =
(a-c * b-c) + a-c =
a-c(b-c + 1) =
(1/ac) * (1/bc + 1) =
1/(ab)c + 1/ac)