Select The Correct Answer.What Is The Solution To This Equation? ( 1 4 ) X + 1 = 32 \left(\frac{1}{4}\right)^{x+1}=32 ( 4 1 ​ ) X + 1 = 32

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Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore the solution to the equation (14)x+1=32\left(\frac{1}{4}\right)^{x+1}=32. We will break down the problem into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is (14)x+1=32\left(\frac{1}{4}\right)^{x+1}=32. To solve this equation, we need to understand the properties of exponents and how to manipulate them.

Exponent Properties

Exponents are a shorthand way of writing repeated multiplication. For example, aba^b can be written as a×a×a×…×aa \times a \times a \times \ldots \times a (b times). In the given equation, we have (14)x+1\left(\frac{1}{4}\right)^{x+1}, which means 14\frac{1}{4} is being multiplied by itself (x+1) times.

Understanding the Base

The base of an exponential expression is the number being raised to a power. In this case, the base is 14\frac{1}{4}. We need to understand the properties of this base to solve the equation.

Manipulating the Equation

To solve the equation, we need to manipulate it to isolate the variable x. We can start by rewriting the equation in a more manageable form.

Rewriting the Equation

We can rewrite the equation as (14)x+1=25\left(\frac{1}{4}\right)^{x+1}=2^5. This is because 32=2532=2^5.

Using Exponent Properties

We can use the property of exponents that states (ab)c=abc(a^b)^c=a^{bc}. We can rewrite the equation as (14)x+1=(12)10\left(\frac{1}{4}\right)^{x+1}=\left(\frac{1}{2}\right)^{10}.

Simplifying the Equation

We can simplify the equation by rewriting 14\frac{1}{4} as (12)2\left(\frac{1}{2}\right)^2. The equation becomes (12)2(x+1)=(12)10\left(\frac{1}{2}\right)^{2(x+1)}=\left(\frac{1}{2}\right)^{10}.

Equating Exponents

Since the bases are the same, we can equate the exponents. We get 2(x+1)=102(x+1)=10.

Solving for x

We can solve for x by subtracting 2 from both sides of the equation. We get x+1=5x+1=5. Subtracting 1 from both sides gives us x=4x=4.

Conclusion

In this article, we solved the equation (14)x+1=32\left(\frac{1}{4}\right)^{x+1}=32 using exponent properties and manipulation. We rewrote the equation in a more manageable form, used exponent properties to simplify it, and equated the exponents to solve for x. The solution to the equation is x=4.

Frequently Asked Questions

Q: What is the solution to the equation (14)x+1=32\left(\frac{1}{4}\right)^{x+1}=32?

A: The solution to the equation is x=4.

Q: How do I solve exponential equations?

A: To solve exponential equations, you need to understand the properties of exponents and how to manipulate them. You can rewrite the equation in a more manageable form, use exponent properties to simplify it, and equate the exponents to solve for the variable.

Q: What are the properties of exponents?

A: The properties of exponents include:

  • (ab)c=abc(a^b)^c=a^{bc}
  • ab×ac=ab+ca^b \times a^c = a^{b+c}
  • abac=ab−c\frac{a^b}{a^c} = a^{b-c}

Additional Resources

Final Thoughts

Solving exponential equations can be challenging, but with the right approach, they can be tackled with ease. By understanding the properties of exponents and how to manipulate them, you can solve equations like (14)x+1=32\left(\frac{1}{4}\right)^{x+1}=32. Remember to rewrite the equation in a more manageable form, use exponent properties to simplify it, and equate the exponents to solve for the variable.

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Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will provide a comprehensive Q&A guide to help you understand and solve 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, 2x2^x is an exponential expression.

Q: How do I solve exponential equations?

A: To solve exponential equations, you need to understand the properties of exponents and how to manipulate them. You can rewrite the equation in a more manageable form, use exponent properties to simplify it, and equate the exponents to solve for the variable.

Q: What are the properties of exponents?

A: The properties of exponents include:

  • (ab)c=abc(a^b)^c=a^{bc}
  • ab×ac=ab+ca^b \times a^c = a^{b+c}
  • abac=ab−c\frac{a^b}{a^c} = a^{b-c}

Q: How do I rewrite an exponential equation in a more manageable form?

A: To rewrite an exponential equation in a more manageable form, you can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Use exponent properties to rewrite the equation in a more manageable form.
  3. Equate the exponents to solve for the variable.

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

A: An exponential equation involves an exponential expression, while a logarithmic equation involves a logarithmic expression. For example, 2x2^x is an exponential equation, while log⁡2(x)\log_2(x) is a logarithmic equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to understand the properties of logarithms and how to manipulate them. You can rewrite the equation in a more manageable form, use logarithmic properties to simplify it, and equate the logarithms to solve for the variable.

Q: What are the properties of logarithms?

A: The properties of logarithms include:

  • log⁡a(b)=c  ⟹  ac=b\log_a(b) = c \implies a^c = b
  • log⁡a(b)+log⁡a(c)=log⁡a(bc)\log_a(b) + \log_a(c) = \log_a(bc)
  • log⁡a(b)−log⁡a(c)=log⁡a(bc)\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})

Q: How do I use a calculator to solve an exponential equation?

A: To use a calculator to solve an exponential equation, you can follow these steps:

  1. Enter the equation into the calculator.
  2. Use the exponent key to raise the base to the power.
  3. Use the equals key to solve for 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 simplifying the equation before solving it.
  • Not using exponent properties to rewrite the equation in a more manageable form.
  • Not equating the exponents to solve for the variable.

Conclusion

In this article, we provided a comprehensive Q&A guide to help you understand and solve exponential equations. We covered topics such as the properties of exponents, rewriting exponential equations in a more manageable form, and using calculators to solve exponential equations. By following these steps and avoiding common mistakes, you can solve exponential equations with ease.

Additional Resources

Final Thoughts

Solving exponential equations can be challenging, but with the right approach, they can be tackled with ease. By understanding the properties of exponents and logarithms, and using calculators to solve equations, you can solve exponential equations with confidence. Remember to simplify the equation before solving it, use exponent properties to rewrite the equation in a more manageable form, and equate the exponents to solve for the variable.