Which Of The Following Is The Solution To The Equation $25^{(z+4)}=125$?A. $z=5.5$ B. $z=3.5$ C. $z=-2.5$ D. $z=-0.5$

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving a specific type of exponential equation, namely the equation $25^{(z+4)}=125$. We will explore the different methods of solving this equation and provide a step-by-step guide to help readers understand the solution.

Understanding Exponential Equations

Exponential equations involve variables raised to a power, and the goal is to solve for the variable. In the given equation, we have $25^{(z+4)}=125$. To solve this equation, we need to understand the properties of exponents and how to manipulate them.

Properties of Exponents

Exponents have several properties that can be used to simplify and solve exponential equations. Some of the key properties include:

  • Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
  • Power of a Power: When raising a power to another power, we multiply the exponents. For example, $(am)n = a^{mn}$.
  • Quotient of Powers: When dividing two powers with the same base, we subtract the exponents. For example, $\frac{am}{an} = a^{m-n}$.

Solving the Equation

To solve the equation $25^{(z+4)}=125$, we can start by rewriting the equation in a more manageable form. We can rewrite 125 as $5^3$ and 25 as $5^2$. This gives us:

52(z+4)=535^{2(z+4)}=5^3

Now, we can use the property of exponents that states when two powers with the same base are equal, their exponents are also equal. This gives us:

2(z+4)=32(z+4)=3

Next, we can simplify the equation by distributing the 2 to the terms inside the parentheses:

2z+8=32z+8=3

Now, we can isolate the variable z by subtracting 8 from both sides of the equation:

2z=−52z=-5

Finally, we can solve for z by dividing both sides of the equation by 2:

z=−52z=-\frac{5}{2}

Conclusion

In this article, we have explored the solution to the equation $25^{(z+4)}=125$. We have used the properties of exponents to simplify the equation and solve for the variable z. The solution to the equation is $z=-\frac{5}{2}$, which corresponds to option D.

Final Answer

The final answer is: −2.5\boxed{-2.5}

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Introduction

In our previous article, we explored the solution to the equation $25^{(z+4)}=125$. We used the properties of exponents to simplify the equation and solve for the variable z. In this article, we will answer some of the most frequently asked questions related to solving exponential equations.

Q&A

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

A: An exponential equation involves a variable raised to a power, while a linear equation involves a variable with a coefficient. For example, $2x+3$ is a linear equation, while $2^x=5$ is an exponential equation.

Q: How do I know which property of exponents to use when solving an exponential equation?

A: When solving an exponential equation, you need to identify the properties of exponents that can be used to simplify the equation. For example, if you have an equation like $a^m \cdot a^n = a^{m+n}$, you can use the product of powers property to simplify the equation.

Q: Can I use the same method to solve all exponential equations?

A: No, not all exponential equations can be solved using the same method. The method used to solve an exponential equation depends on the specific equation and the properties of exponents that can be used to simplify it.

Q: What if I have an exponential equation with a negative exponent?

A: If you have an exponential equation with a negative exponent, you can use the property of exponents that states $a^{-n} = \frac{1}{a^n}$. For example, if you have the equation $2^{-x}=3$, you can rewrite it as $\frac{1}{2^x}=3$.

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations. However, it's always a good idea to check your work by plugging the solution back into the original equation to make sure it's true.

Q: What if I have an exponential equation with a variable in the exponent?

A: If you have an exponential equation with a variable in the exponent, you can use the property of exponents that states $a{m+n}=am \cdot a^n$. For example, if you have the equation $2^{x+2}=5$, you can rewrite it as $2^x \cdot 2^2=5$.

Conclusion

In this article, we have answered some of the most frequently asked questions related to solving exponential equations. We have covered topics such as the difference between exponential and linear equations, how to identify the properties of exponents to use when solving an exponential equation, and how to handle negative exponents and variables in the exponent.

Final Tips

  • Always read the problem carefully and identify the properties of exponents that can be used to simplify the equation.
  • Use a calculator to check your work and make sure the solution is true.
  • Practice solving exponential equations to become more comfortable with the properties of exponents and how to apply them to solve equations.

Final Answer

The final answer is: −2.5\boxed{-2.5}