If 3 X + 8 = 27 2 X + 1 3^{x+8} = 27^{2x+1} 3 X + 8 = 2 7 2 X + 1 , Then The Value Of X X X Will Be:(a) 1 (b) -2 (c) 3 (d) -1 18 Men Can Do A Piece Of Work In 20 Days. The Number Of Days Needed To Complete The Job If 24 Men Are Employed To Do The Same Piece Of Work Is:(a)

Introduction

In this article, we will explore two different mathematical problems: solving exponential equations and work problems. We will start by solving the exponential equation $3^{x+8} = 27^{2x+1}$, and then move on to the work problem involving 18 men and 24 men.

Solving Exponential Equations

Exponential equations are equations that involve exponential expressions. These equations can be solved using various methods, including the use of logarithms. In this section, we will solve the exponential equation $3^{x+8} = 27^{2x+1}$.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it. We can rewrite $27$ as $3^3$, so the equation becomes:

$$3^{x+8} = (3^3)^{2x+1}$$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify the right-hand side of the equation:

$$3^{x+8} = 3^{3(2x+1)}$$

Step 2: Equate the Exponents

Since the bases of the two exponential expressions are the same, we can equate the exponents:

$$x+8 = 3(2x+1)$$

Step 3: Solve for x

Now, we can solve for $x$ by simplifying the equation:

$$x+8 = 6x+3$$

Subtracting $x$ from both sides gives:

$$8 = 5x+3$$

Subtracting $3$ from both sides gives:

$$5 = 5x$$

Dividing both sides by $5$ gives:

$$x = 1$$

Therefore, the value of $x$ is $1$.

Work Problems

Work problems involve finding the time it takes to complete a job when a certain number of people are working on it. In this section, we will solve the work problem involving 18 men and 24 men.

Step 1: Understand the Problem

We are given that 18 men can complete a piece of work in 20 days. We need to find the number of days needed to complete the same job if 24 men are employed to do the same piece of work.

Step 2: Use the Concept of Work Done

The work done by a group of people is directly proportional to the number of people working and the time they work. Since the work done is the same in both cases, we can set up a proportion:

$$\frac{18 \text{ men} \times 20 \text{ days}}{1 \text{ job}} = \frac{24 \text{ men} \times x \text{ days}}{1 \text{ job}}$$

Step 3: Solve for x

Now, we can solve for $x$ by simplifying the proportion:

$$18 \times 20 = 24 \times x$$

$$360 = 24x$$

Dividing both sides by $24$ gives:

$$x = 15$$

Therefore, the number of days needed to complete the job if 24 men are employed to do the same piece of work is $15$.

Conclusion

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

A: An exponential equation is an equation that involves exponential expressions, where the variable is raised to a power. A linear equation, on the other hand, is an equation that involves a linear expression, where the variable is not raised to a power.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use various methods, including the use of logarithms. You can also try to simplify the equation by rewriting the exponential expressions in a different form.

Q: What is the concept of work done in work problems?

A: The work done by a group of people is directly proportional to the number of people working and the time they work. This means that if you increase the number of people working or the time they work, the work done will also increase.

Q: How do I solve a work problem?

A: To solve a work problem, you can use the concept of work done. You can set up a proportion to relate the work done by the original group of people to the work done by the new group of people.

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

A: A proportion is a statement that two ratios are equal, while an equation is a statement that two expressions are equal. In a proportion, the ratios are often expressed as fractions, while in an equation, the expressions are often expressed as equalities.

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

A: To use logarithms to solve an exponential equation, you can take the logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation and solve for the variable.

Q: What is the significance of the base of an exponential expression?

A: The base of an exponential expression is the number that is raised to a power. The base can affect the growth rate of the exponential expression, and can also affect the difficulty of solving the equation.

Q: How do I choose the correct method to solve an exponential equation?

A: To choose the correct method to solve an exponential equation, you should consider the complexity of the equation and the tools you have available. If the equation is simple, you may be able to solve it using basic algebraic manipulations. If the equation is more complex, you may need to use logarithms or other advanced techniques.

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 the correct method to solve the equation
• Not checking the solution to make sure it is valid
• Not considering the domain of the equation

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

A: To check your solution to an exponential equation, you should plug the solution back into the original equation and make sure it is true. You should also consider the domain of the equation and make sure the solution is valid.

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

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

• Modeling population growth
• Modeling financial growth
• Modeling chemical reactions
• Modeling electrical circuits

Q: How do I use exponential equations to model real-world problems?

A: To use exponential equations to model real-world problems, you should first identify the key variables and parameters in the problem. You should then use the exponential equation to model the relationship between the variables and parameters. Finally, you should use the equation to make predictions or solve problems.