How to Solve Exponential Equations with Division?

To solve exponential equations involving division, the key is to remember the rules of exponents, especially when the bases are the same. Essentially, when you divide exponential expressions with the same base, you subtract the exponents.

Here’s a breakdown:

Dividing Exponential Expressions with the Same Base

The primary rule to remember is:

x^m / xⁿ = x^(m-n)

Where:

• x is the base (must be the same for both expressions).
• m and n are the exponents.

According to our reference material, "in order to divide one exponential expression by another with the same base, you just need to subtract the exponents."

Steps to Solve

1. Ensure the Bases are the Same: This method only works if the bases of the exponential expressions are identical. If they're not, you might need to manipulate the equation to achieve a common base or resort to other methods like logarithms (which are beyond the scope of this explanation).
2. Subtract the Exponents: Subtract the exponent in the denominator from the exponent in the numerator. The result becomes the new exponent of the common base.
3. Simplify: Simplify the resulting expression.

Examples

Here are some examples to illustrate the process:

• Example 1: Simplify 2⁵ / 2²

  • Bases are the same (both are 2).
  • Subtract the exponents: 5 - 2 = 3
  • Result: 2³ = 8

• Example 2: Simplify 5^-3 / 5^-5

  • Bases are the the same (both are 5).
  • Subtract the exponents: -3 - (-5) = -3 + 5 = 2
  • Result: 5² = 25

• Example 3: Solve for x: 3^x+2 / 3^x = 9

  • Bases are the same (both are 3).
  • Subtract the exponents: (x + 2) - x = 2
  • Simplified equation: 3² = 9
  • Since 3² = 9, the equation holds true. This example shows how the rule applies within an equation.

When Bases Are Different

Our reference states: "If the terms have different bases, there is not much that can be done to simplify the expression."

If you encounter division with different bases, you cannot directly subtract the exponents. In such cases, you might need to:

• Simplify each expression separately: Evaluate each exponential term individually if possible.
• Use logarithms: Logarithms can sometimes help simplify or solve equations with different bases, but this involves a more advanced technique.
• Rewrite the bases: Attempt to express one or both bases in terms of a common base. This isn't always possible, but sometimes it can simplify the problem.