How to Divide Exponents with Different Bases?

Dividing exponents with different bases is not straightforward and doesn't have a single, simple rule like when the bases are the same.

Here's a breakdown:

Understanding the Challenge

• When you divide exponents with the same base, you simply subtract the exponents (e.g., x⁴ / x² = x^(4-2) = x²), as mentioned in the reference.
• However, when the bases are different (e.g., x⁴ / y²), this simple subtraction rule does not apply.

What To Do With Different Bases:

1. No Direct Simplification: In most cases, you cannot directly simplify an expression with different bases using exponent rules for division.
2. Keep Separate: Expressions like x⁴ / y² are typically left as is.
3. Look for Opportunities to Simplify Bases: The best course of action involves:
   • Factoring: See if you can break down the bases into common factors and then cancel out common factors. If there are common factors, you may be able to change bases or simplify. For example if you had 6²/3², you could rewrite it as (3*2)²/3², which simplifies to 3²*2²/3² and allows us to remove the 3².
   • Rewriting bases: Sometimes it will be necessary to write one base using a new base. For instance if we have 4²/2³, we can rewrite 4 as 2² giving us (2²)²/2³ which simplifies to 2⁴/2³ or 2.
   • Converting to fractions or decimals: Sometimes it may be helpful to evaluate both the top and bottom separately and convert to decimal or fraction form. This is not simplification using exponent rules, but is a calculation that is sometimes helpful.
4. Evaluate separately:
   • You can evaluate the top and bottom separately if you have a specific value for the variables.

Example

Consider the expression: a³ / b²

• You cannot directly simplify this expression because the bases are 'a' and 'b,' which are different.
• Unless there is more information to relate a and b you should leave the expression as is.

When It Might be Possible

If there's a relationship between the bases, such as:

• Common factor: If a and b had a common factor that could be factored out of both bases, you might be able to simplify further.
• Variable Substitution: If a = b² you could replace a with b².

Summary

• Dividing exponents with different bases typically does not lead to simple simplifications.
• The primary focus should be on simplifying the bases if possible through factoring or rewriting bases.
• If no simplification of bases is possible the expressions are usually left as is.