How To Solve 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.
\(\begin{aligned}
6^{24}&=6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3 × 6^3 \
&= 216 × 216 × 216 × 216 × 216 × 216 × 216 × 216 \
&= 46656 × 46656 × 46656 × 46656 \
&= 4.738 × 10^{18}
\end{aligned}\)
Things Needed
- Pen or pencil
- Paper
TL;DR (Too Long; Didn't Read)
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
Cite This Article
MLA
McKenzie, Grant D.. "How To Solve Large Exponents" sciencing.com, https://www.sciencing.com/solve-large-exponents-5886569/. 16 November 2020.
APA
McKenzie, Grant D.. (2020, November 16). How To Solve Large Exponents. sciencing.com. Retrieved from https://www.sciencing.com/solve-large-exponents-5886569/
Chicago
McKenzie, Grant D.. How To Solve Large Exponents last modified March 24, 2022. https://www.sciencing.com/solve-large-exponents-5886569/