Power Rule

Find the prime factors of the exponent. Example: 6²⁴

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

Use the power rule for exponents to set up the problem. The power rule states:

(x^a)^b = x^{(a × b)}

So

6^{24} = 6^{(2 × 2 × 2 × 3)} = (((6^2)^2)^2)^3

Solve the problem from the inside out.

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

Product Rule

Break the exponent down into a sum. Make sure the components are small enough to work with as exponents and do not include 1 or 0.

Example: 6²⁴

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

Use the product rule of exponents to set up the problem. The product rule states:

x^a × x^b = x^{a+b}

So

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

Solve the problem.

\begin{aligned} 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} \end{aligned}

Things You'll Need

• Pen or pencil
• Paper

Tips

• For some problems, a combination of both techniques may make the problem easier. For example: ​x​²¹ = (​x​⁷)³ (power rule), and ​x​⁷ = ​x​³ × ​x​² × ​x​² (product rule). Combining the two, you get: ​x​²¹ = (​x​³ × ​x​² × ​x​²)³