Solve For $z$.$6^2 = \frac{6^z}{6^3}$A. $z = 2$ B. $z = 4$ C. $z = 5$ D. $z = 3$

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving a specific type of exponential equation, where the variable is part of the exponent. We will use the given equation 62=6z636^2 = \frac{6^z}{6^3} as a case study to demonstrate the step-by-step process of solving exponential equations.

Understanding Exponential Equations

Exponential equations involve variables that are raised to a power. In the given equation, 62=6z636^2 = \frac{6^z}{6^3}, the variable zz is part of the exponent. To solve this equation, we need to isolate the variable zz and find its value.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by getting rid of the fraction. We can do this by multiplying both sides of the equation by 636^3. This will eliminate the fraction and make it easier to work with.

62⋅63=6z63⋅636^2 \cdot 6^3 = \frac{6^z}{6^3} \cdot 6^3

Using the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}, we can simplify the left-hand side of the equation.

62+3=6z6^{2+3} = 6^z

65=6z6^5 = 6^z

Step 2: Equate the Exponents

Now that we have simplified the equation, we can equate the exponents on both sides. Since the bases are the same (6), we can set the exponents equal to each other.

5=z5 = z

Conclusion

In this article, we have demonstrated the step-by-step process of solving an exponential equation. We started by simplifying the equation, and then equated the exponents on both sides. The final answer is z=5z = 5.

Answer Key

The correct answer is:

  • A. z=2z = 2 (Incorrect)
  • B. z=4z = 4 (Incorrect)
  • C. z=5z = 5 (Correct)
  • D. z=3z = 3 (Incorrect)

Tips and Tricks

When solving exponential equations, it's essential to remember the following tips and tricks:

  • Simplify the equation by getting rid of fractions and combining like terms.
  • Use the rule of exponents to simplify expressions.
  • Equate the exponents on both sides of the equation.
  • Check your answer by plugging it back into the original equation.

By following these tips and tricks, you'll be able to solve exponential equations with ease and confidence.

Real-World Applications

Exponential equations have numerous real-world applications in fields such as finance, science, and engineering. For example, compound interest is calculated using exponential equations, and population growth can be modeled using exponential functions.

Conclusion

In conclusion, solving exponential equations requires a deep understanding of the underlying principles and a step-by-step approach. By simplifying the equation, equating the exponents, and checking the answer, you'll be able to solve exponential equations with ease and confidence. Remember to use the tips and tricks outlined in this article to help you solve exponential equations in the future.

Frequently Asked Questions

Q: What is an exponential equation? A: An exponential equation is an equation that involves variables that are raised to a power.

Q: How do I simplify an exponential equation? A: To simplify an exponential equation, get rid of fractions and combine like terms.

Q: How do I equate the exponents on both sides of an exponential equation? A: Since the bases are the same, you can set the exponents equal to each other.

Q: What are some real-world applications of exponential equations? A: Exponential equations have numerous real-world applications in fields such as finance, science, and engineering.

Glossary

  • Exponential equation: An equation that involves variables that are raised to a power.
  • Exponent: A number that represents the power to which a base is raised.
  • Base: The number that is raised to a power.
  • Rule of exponents: A set of rules that govern the behavior of exponents when they are combined or simplified.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by Michael Spivak
  • [3] "Exponential Functions" by Wolfram MathWorld

About the Author

Introduction

Exponential equations can be a challenging topic for many students, but with the right guidance and practice, they can become a breeze. In this article, we will answer some of the most frequently asked questions about exponential equations, providing you with a deeper understanding of this important mathematical concept.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves variables that are raised to a power. For example, the equation 2x=162^x = 16 is an exponential equation because the variable xx is raised to a power.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, get rid of fractions and combine like terms. For example, the equation 2322=2x\frac{2^3}{2^2} = 2^x can be simplified by combining the exponents: 23−2=2x2^{3-2} = 2^x, which simplifies to 21=2x2^1 = 2^x.

Q: How do I equate the exponents on both sides of an exponential equation?

A: Since the bases are the same, you can set the exponents equal to each other. For example, in the equation 2x=232^x = 2^3, you can equate the exponents: x=3x = 3.

Q: What is the rule of exponents?

A: The rule of exponents is a set of rules that govern the behavior of exponents when they are combined or simplified. The main rules are:

  • am⋅an=am+na^m \cdot a^n = a^{m+n}
  • aman=am−n\frac{a^m}{a^n} = a^{m-n}
  • (am)n=amn(a^m)^n = a^{mn}

Q: How do I solve an exponential equation with a variable in the exponent?

A: To solve an exponential equation with a variable in the exponent, you need to isolate the variable. For example, in the equation 22x=162^{2x} = 16, you can start by taking the logarithm of both sides: log⁡(22x)=log⁡(16)\log(2^{2x}) = \log(16). Then, you can use the property of logarithms that states log⁡(ab)=blog⁡(a)\log(a^b) = b \log(a) to simplify the equation: 2xlog⁡(2)=log⁡(16)2x \log(2) = \log(16). Finally, you can solve for xx by dividing both sides by 2log⁡(2)2 \log(2).

Q: What are some real-world applications of exponential equations?

A: Exponential equations have numerous real-world applications in fields such as finance, science, and engineering. For example, compound interest is calculated using exponential equations, and population growth can be modeled using exponential functions.

Q: How do I check my answer to an exponential equation?

A: To check your answer to an exponential equation, plug it back into the original equation and simplify. If the equation holds true, then your answer is correct.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

  • Not simplifying the equation before solving
  • Not equating the exponents on both sides of the equation
  • Not checking the answer by plugging it back into the original equation

Conclusion

In conclusion, exponential equations can be a challenging topic, but with the right guidance and practice, they can become a breeze. By understanding the basics of exponential equations, including simplifying, equating exponents, and checking answers, you can solve these equations with confidence. Remember to avoid common mistakes and to practice regularly to become proficient in solving exponential equations.

Glossary

  • Exponential equation: An equation that involves variables that are raised to a power.
  • Exponent: A number that represents the power to which a base is raised.
  • Base: The number that is raised to a power.
  • Rule of exponents: A set of rules that govern the behavior of exponents when they are combined or simplified.
  • Logarithm: The inverse operation of exponentiation.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by Michael Spivak
  • [3] "Exponential Functions" by Wolfram MathWorld

About the Author

The author is a mathematics educator with a passion for teaching and learning. They have a strong background in mathematics and have taught a variety of courses, including algebra, calculus, and statistics.