Strategies for Solving Large Exponents

As with most problems in basic algebra, solving large exponents requires factoring. If you factor the exponent down until all the factors are prime numbers – a process called prime factorization – you can then apply the power rule of exponents to solve the problem. Additionally, you can break the exponent down by addition rather than multiplication and apply the product rule for exponents to solve the problem. A little practice will help you predict which method will be easiest for the problem you are faced with.

Power Rule

1. Find Prime Factors

Find the prime factors of the exponent. Example: 624

(24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3)

2. Apply the Power Rule

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)

3. Calculate the Exponents

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

1. Deconstruct the Exponent

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)

2. Apply the Product Rule

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)

3. Compute the Exponents

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})