Which Equation Can Be Used To Find The Solution Of $\left(\frac{1}{3}\right)^{d-5}=81$?A. $-d-5=4$B. $d-5=4$C. $-d+5=4$D. $d+5=4$

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the underlying principles. In this article, we will focus on solving the equation (13)d−5=81\left(\frac{1}{3}\right)^{d-5}=81. This equation involves an exponential term with a base of 13\frac{1}{3} and an exponent of d−5d-5. Our goal is to isolate the variable dd and find its value.

Understanding Exponential Equations

Exponential equations have the form ax=ba^x=b, where aa is the base, xx is the exponent, and bb is the result. To solve these equations, we need to isolate the exponent xx. In our given equation, the base is 13\frac{1}{3}, and the exponent is d−5d-5. We are given that the result is 8181.

Rewriting the Equation

To solve the equation, we need to rewrite it in a more manageable form. We can start by expressing 8181 as a power of 33, since 81=3481=3^4. This gives us:

(13)d−5=34\left(\frac{1}{3}\right)^{d-5}=3^4

Using Exponent Properties

Now that we have rewritten the equation, we can use exponent properties to simplify it. Specifically, we can use the property that states (1a)x=a−x\left(\frac{1}{a}\right)^x=a^{-x}. Applying this property to our equation, we get:

3−(d−5)=343^{-(d-5)}=3^4

Simplifying the Equation

We can simplify the equation by combining the exponents. Since the bases are the same, we can add the exponents:

−d+5=4-d+5=4

Solving for dd

Now that we have simplified the equation, we can solve for dd. To do this, we need to isolate the variable dd. We can do this by subtracting 55 from both sides of the equation:

−d=−1-d=-1

Final Answer

Multiplying both sides of the equation by −1-1, we get:

d=1d=1

Conclusion

In this article, we have solved the equation (13)d−5=81\left(\frac{1}{3}\right)^{d-5}=81 using exponent properties and simplification techniques. We have shown that the correct equation to use is −d+5=4-d+5=4, and we have solved for the value of dd. The final answer is d=1d=1.

Answer Key

The correct answer is C. −d+5=4-d+5=4.

Additional Tips and Tricks

  • When solving exponential equations, it's essential to use exponent properties to simplify the equation.
  • Make sure to isolate the variable by adding or subtracting the same value from both sides of the equation.
  • Use the property that states (1a)x=a−x\left(\frac{1}{a}\right)^x=a^{-x} to simplify the equation.

Real-World Applications

Exponential equations have numerous real-world applications, including:

  • Modeling population growth and decline
  • Calculating compound interest
  • Analyzing chemical reactions
  • Solving problems in physics and engineering

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid common mistakes such as:

  • Not using exponent properties to simplify the equation
  • Not isolating the variable by adding or subtracting the same value from both sides of the equation
  • Not checking the solution to ensure it satisfies the original equation

Conclusion

Solving exponential equations requires a clear understanding of exponent properties and simplification techniques. By following the steps outlined in this article, you can solve equations like (13)d−5=81\left(\frac{1}{3}\right)^{d-5}=81 and find the value of the variable dd. Remember to use exponent properties, simplify the equation, and isolate the variable to ensure a correct solution.

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential term, which is a number raised to a power. For example, the equation (13)d−5=81\left(\frac{1}{3}\right)^{d-5}=81 is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable by using exponent properties and simplification techniques. This involves rewriting the equation in a more manageable form, using exponent properties to simplify it, and isolating the variable.

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 exponent properties to simplify the equation
  • Not isolating the variable by adding or subtracting the same value from both sides of the equation
  • Not checking the solution to ensure it satisfies the original equation

Q: How do I use exponent properties to simplify an exponential equation?

A: Exponent properties can be used to simplify an exponential equation by combining the exponents or rewriting the equation in a more manageable form. For example, the property (1a)x=a−x\left(\frac{1}{a}\right)^x=a^{-x} can be used to rewrite the equation (13)d−5=81\left(\frac{1}{3}\right)^{d-5}=81 as 3−(d−5)=343^{-(d-5)}=3^4.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation involves an exponential term, which is a number raised to a power, while a linear equation involves a linear term, which is a number multiplied by a variable. For example, the equation 2x+3=52x+3=5 is a linear equation, while the equation (13)d−5=81\left(\frac{1}{3}\right)^{d-5}=81 is an exponential equation.

Q: Can exponential equations be solved using algebraic methods?

A: Yes, exponential equations can be solved using algebraic methods, such as isolating the variable and using exponent properties to simplify the equation. However, exponential equations often require a different approach than linear equations.

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

A: Exponential equations have numerous real-world applications, including:

  • Modeling population growth and decline
  • Calculating compound interest
  • Analyzing chemical reactions
  • Solving problems in physics and engineering

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

A: To check your solution to an exponential equation, you need to substitute the value of the variable back into the original equation and verify that it is true. For example, if you solve the equation (13)d−5=81\left(\frac{1}{3}\right)^{d-5}=81 and get d=1d=1, you need to substitute d=1d=1 back into the original equation and verify that it is true.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

  • Using exponent properties to simplify the equation
  • Isolating the variable by adding or subtracting the same value from both sides of the equation
  • Checking the solution to ensure it satisfies the original equation
  • Using a calculator to check your solution

Conclusion

Solving exponential equations requires a clear understanding of exponent properties and simplification techniques. By following the steps outlined in this article and avoiding common mistakes, you can solve exponential equations and find the value of the variable. Remember to use exponent properties, simplify the equation, and isolate the variable to ensure a correct solution.