What Is The Value Of $m$?$18^m \cdot 18^5 = \left(18^3\right)^4$A. \$m = 12$[/tex\]B. $m = 1$C. $m = 7$D. \$m = 0$[/tex\]

by ADMIN 132 views

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will explore how to solve exponential equations, with a focus on the given problem: $18^m \cdot 18^5 = \left(183\right)4$. We will break down the solution step by step, using the properties of exponents to simplify the equation and find the value of $m$.

Understanding Exponents

Before we dive into the solution, let's review the basics of exponents. An exponent is a small number that is raised to the power of a larger number. For example, $2^3$ means $2$ multiplied by itself $3$ times: $2 \times 2 \times 2 = 8$. The exponent $3$ tells us how many times to multiply the base $2$.

Simplifying the Equation

Now that we have a basic understanding of exponents, let's simplify the given equation: $18^m \cdot 18^5 = \left(183\right)4$. We can start by using the property of exponents that states: $a^m \cdot a^n = a^{m+n}$. This means that when we multiply two numbers with the same base, we can add their exponents.

Applying this property to the given equation, we get: $18^m \cdot 18^5 = 18^m+5}$. Now, let's look at the right-hand side of the equation $\left(183\right)4$. We can use the property of exponents that states: $(am)n = a^{mn$. This means that when we raise a number with an exponent to another power, we can multiply the exponents.

Applying this property to the right-hand side of the equation, we get: $\left(183\right)4 = 18^3 \cdot 4} = 18^{12}$. Now, we can equate the two sides of the equation $18^{m+5 = 18^{12}$.

Equating Exponents

Since the bases are the same, we can equate the exponents: $m+5 = 12$. Now, we can solve for $m$ by subtracting $5$ from both sides of the equation: $m = 12 - 5$.

Solving for $m$

m = 7$. **Conclusion** -------------- In this article, we have solved the exponential equation: $18^m \cdot 18^5 = \left(18^3\right)^4$. We used the properties of exponents to simplify the equation and find the value of $m$. The final answer is: $m = 7$. **Final Answer** ---------------- The final answer is: **C. $m = 7$**. **Additional Resources** ------------------------- If you are struggling with exponential equations or need additional practice, here are some additional resources: * Khan Academy: Exponents and Exponential Functions * Mathway: Exponential Equations Solver * Wolfram Alpha: Exponential Equations Calculator **Common Mistakes** ------------------- When solving exponential equations, it's easy to make mistakes. Here are some common mistakes to avoid: * Not using the properties of exponents correctly * Not equating the exponents when the bases are the same * Not solving for the variable correctly **Tips and Tricks** ------------------- Here are some tips and tricks to help you solve exponential equations: * Use the properties of exponents to simplify the equation * Equate the exponents when the bases are the same * Solve for the variable correctly by using inverse operations **Real-World Applications** ------------------------- Exponential equations have many real-world applications, including: * Modeling population growth * Calculating compound interest * Analyzing data in science and engineering **Conclusion** -------------- In conclusion, solving exponential equations requires a deep understanding of the properties of exponents. By using the properties of exponents to simplify the equation and equating the exponents when the bases are the same, we can solve for the variable and find the value of $m$. The final answer is: $m = 7$.<br/> **Exponential Equations Q&A** ========================== **Q: What is an exponential equation?** -------------------------------------- A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $2^3$ is an exponential expression, where $2$ is the base and $3$ is the exponent. **Q: How do I simplify an exponential equation?** ---------------------------------------------- A: To simplify an exponential equation, you can use the properties of exponents. For example, if you have $a^m \cdot a^n$, you can add the exponents to get $a^{m+n}$. You can also use the property $(a^m)^n = a^{mn}$ to simplify expressions with multiple exponents. **Q: How do I solve an exponential equation?** ---------------------------------------------- A: To solve an exponential equation, you need to isolate the variable. This can be done by using inverse operations, such as dividing or taking the logarithm of both sides of the equation. For example, if you have $2^x = 8$, you can take the logarithm of both sides to get $x = \log_2(8)$. **Q: What is the difference between an exponential equation and a linear equation?** -------------------------------------------------------------------------------- A: An exponential equation is an equation that involves an exponential expression, while a linear equation is an equation that involves a linear expression, such as $2x + 3 = 5$. Exponential equations can have multiple solutions, while linear equations typically have a single solution. **Q: Can I use a calculator to solve an exponential equation?** --------------------------------------------------------- A: Yes, you can use a calculator to solve an exponential equation. However, it's always a good idea to check your work by plugging the solution back into the original equation. **Q: How do I graph an exponential function?** ---------------------------------------------- A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function. **Q: What are some common mistakes to avoid when solving exponential equations?** -------------------------------------------------------------------------------- A: Some common mistakes to avoid when solving exponential equations include: * Not using the properties of exponents correctly * Not equating the exponents when the bases are the same * Not solving for the variable correctly by using inverse operations **Q: How do I apply exponential equations to real-world problems?** ---------------------------------------------------------------- A: Exponential equations can be used to model real-world problems, such as population growth, compound interest, and data analysis. For example, you can use an exponential equation to model the growth of a population over time. **Q: What are some examples of exponential equations in real-world applications?** -------------------------------------------------------------------------------- A: Some examples of exponential equations in real-world applications include: * Modeling population growth: $P(t) = P_0 \cdot e^{rt}

  • Calculating compound interest: $A = P \cdot (1 + r)^t$
  • Analyzing data in science and engineering: $y = a \cdot e^{bx}$

Q: How do I choose the correct base for an exponential equation?

A: The base of an exponential equation depends on the problem you are trying to solve. For example, if you are modeling population growth, you may use a base of $e$, while if you are calculating compound interest, you may use a base of $1 + r$.

Q: What are some common bases used in exponential equations?

A: Some common bases used in exponential equations include:

  • e$ (the base of the natural logarithm)

  • 2$ (the base of the binary logarithm)

  • 10$ (the base of the common logarithm)

  • 1 + r$ (the base of the compound interest formula)

Conclusion

In conclusion, exponential equations are a powerful tool for modeling real-world problems. By understanding the properties of exponents and how to solve exponential equations, you can apply these concepts to a wide range of applications. Remember to use the properties of exponents correctly, equate the exponents when the bases are the same, and solve for the variable correctly by using inverse operations.