📚 Topic Summary
Exponential equations are equations where the variable appears in the exponent. The key to solving simple exponential equations is to manipulate the equation so that both sides have the same base. Once the bases are the same, you can set the exponents equal to each other and solve for the variable. For example, if you have $2^x = 2^3$, then $x=3$. Sometimes, you need to rewrite one or both sides of the equation to have a common base before you can solve it. Let's work through some examples!
🧠 Part A: Vocabulary
Match the following terms with their correct definitions:
- Term: Exponent
- Term: Base
- Term: Exponential Equation
- Term: Variable
- Term: Solution
Definitions:
- A symbol (usually a letter) representing an unknown number.
- The value that satisfies the equation.
- The number that is raised to a power.
- An equation where the variable appears in the exponent.
- The power to which a number is raised.
Match the term to the correct definitions (e.g., 1-A, 2-B, etc.)
📝 Part B: Fill in the Blanks
Complete the following paragraph with the correct words:
To solve exponential equations, we aim to have the same ______ on both sides of the equation. Once the bases are equal, we can set the ______ equal to each other. Sometimes, we need to ______ the equation to achieve a common base. For example, if we have $4^x = 8$, we can rewrite both sides with a base of ______. This gives us $(2^2)^x = 2^3$, which simplifies to $2^{2x} = 2^3$. Therefore, $2x=3$, and $x=$ ______.
Word Bank: exponents, rewrite, base, 2, $\frac{3}{2}$
💡 Part C: Critical Thinking
Explain in your own words why it is important to have the same base when solving exponential equations. Give an example to illustrate your explanation.