How to Solve Large Exponents

How to Solve Large Exponents
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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

    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: ​x21 = (​x7)3 (power rule), and ​x7 = ​x3 × ​x2 × ​x2 (product rule). Combining the two, you get: ​x21 = (​x3 × ​x2 × ​x2)3