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When dealing with huge exponents, the key to a clear solution is to break the problem down using factorization. By reducing the exponent to its prime components, you can then apply the power rule of exponents. Alternatively, if the exponent can be expressed as a sum of smaller integers, the product rule offers a simpler path. With a few practice problems, you’ll be able to choose the most efficient strategy for any situation.

Power Rule Method

1. Find the Prime Factors of the Exponent

For example, consider the exponent 24:

24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3

2. Apply the Power Rule

The power rule states that (x^a)^b = x^{a\times b}. Thus:

6^{24} = 6^{(2\times2\times2\times3)} = (((6^2)^2)^2)^3

3. Compute from Inside Out

Step by step:

(((((6^2)^2)^2)^3)
= ((36^2)^2)^3
= (1296^2)^3
= 1679616^3
= 4.738 × 10^{18}

Product Rule Method

1. Deconstruct the Exponent into a Sum

Rewrite 24 as a sum of small, non‑trivial integers, e.g.:

24 = 12 + 12 = 6 + 6 + 6 + 6 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3

2. Apply the Product Rule

The product rule says x^a × x^b = x^{a+b}. Therefore:

6^{24} = 6^{(3+3+3+3+3+3+3+3)} = 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3

3. Compute the Result

6^{24}
= 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3
= 216 × 216 × 216 × 216 × 216 × 216 × 216 × 216
= 46656 × 46656 × 46656 × 46656
= 4.738 × 10^{18}