How to Convert Exponents With Different Bases

In high school math, you convert exponents with different bases to like bases in order to simplify math expressions. For example, to evaluate the expression 2^5*4^2 without the aid of a calculator, one way is to first rewrite each exponent using the same base. Another application of this useful skill is to solve exponential equations with different bases. For instance, to solve the equation 3^(x+2)=27, you need to first write each side as a power of the same base. The process to convert exponents with different bases is straightforward.

Instructions

1. Find a common base. For example, in the expression 9^2*27^2, the bases are 9 and 27. Since these two numbers are both powers of 3, the common base is 3.

2. Express each base in terms of the common base. Continuing the example, 9=3^2 and 27=3^3.

3. Rewrite each power in terms of the common base. Thus, 9^2=(3^2)^2 and 27^2=(3^3)^2.

4. Simplify each power using the rules of exponents. The key rule to use is (a^x)^y=a^(x*y). Thus, 9^2=(3^2)^2=3^4 and 27^2=(3^3)^2=3^6.

5. Combine powers with equal bases. The final answer is 9^2*27^2=3^4*3^6=3^(4+6) or 3^10, which is equal to 59,049.

Tips & Warnings

• Use negative exponents for fractional bases. For instance, 1/4=4^(-1)=2^(-2).

References

• "Algebra 2 and Trigonometry"; Ann Xavier Gantert; 2009