Solve The Given Equation.${ \left(6 {\frac{x}{6}}\right)\left(6 {\frac{x}{5}}\right)=6^8 }$ { x = \, \square \, \text{(Type An Integer Or A Simplified Fraction.)} \}

Mar 13, 2025 by ADMIN 170 views

**Solving Exponential Equations: A Step-by-Step Guide**

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving a specific exponential equation, which involves the product of two exponential expressions equal to a third exponential expression. We will use the properties of exponents to simplify the equation and solve for the variable.

The Given Equation

The given equation is:

\left(6^{\frac{x}{6}}\right)\left(6^{\frac{x}{5}}\right)=6^8

This equation involves the product of two exponential expressions, both of which have a base of 6. The exponents of the two expressions are $\frac{x}{6}$ and $\frac{x}{5}$ , respectively. The equation states that the product of these two expressions is equal to $6^8$ .

Using the Product of Powers Property

To solve this equation, we can use the product of powers property, which states that when we multiply two exponential expressions with the same base, we can add their exponents. In this case, we can add the exponents $\frac{x}{6}$ and $\frac{x}{5}$ to get a single exponent.

Using the product of powers property, we can rewrite the equation as:

6^{\frac{x}{6} + \frac{x}{5}} = 6^8

Simplifying the Exponent

Now that we have a single exponent on the left-hand side of the equation, we can simplify it by finding a common denominator. The least common multiple of 6 and 5 is 30, so we can rewrite the exponent as:

\frac{x}{6} + \frac{x}{5} = \frac{5x + 6x}{30} = \frac{11x}{30}

Therefore, the equation becomes:

6^{\frac{11x}{30}} = 6^8

Equating Exponents

Since the bases of the two exponential expressions are the same (6), we can equate their exponents. This gives us:

\frac{11x}{30} = 8

Solving for x

To solve for x, we can multiply both sides of the equation by 30 to eliminate the fraction:

11x = 8 \times 30

11x = 240

Now, we can divide both sides of the equation by 11 to solve for x:

x = \frac{240}{11}

Conclusion

In this article, we solved an exponential equation involving the product of two exponential expressions. We used the product of powers property to simplify the equation and then equated the exponents to solve for the variable. The final solution is x = $\frac{240}{11}$ .

Discussion

This equation is a classic example of an exponential equation, and solving it requires a deep understanding of exponential functions and their properties. The product of powers property is a powerful tool for simplifying exponential expressions, and it is essential to understand how to apply it in different situations.

Exercises

Solve the equation $2^x \cdot 2^y = 2^{10}$ .
Solve the equation $3^x \cdot 3^y = 3^{12}$ .
Solve the equation $4^x \cdot 4^y = 4^{15}$ .

Answer Key

x = 5, y = 5
x = 4, y = 8
x = 3, y = 12
Frequently Asked Questions: Solving Exponential Equations =============================================================

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a mathematical expression that represents a quantity that grows or decays at a constant rate. Exponential equations often involve variables in the exponent, and solving them requires a deep understanding of exponential functions and their properties.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the product of powers property, which states that when you multiply two exponential expressions with the same base, you can add their exponents. You can also use the quotient of powers property, which states that when you divide two exponential expressions with the same base, you can subtract their exponents.

Q: What is the product of powers property?

A: The product of powers property is a mathematical property that states that when you multiply two exponential expressions with the same base, you can add their exponents. For example, if you have the equation $a^m \cdot a^n = a^{m+n}$ , then the product of powers property allows you to simplify the equation by adding the exponents.

Q: What is the quotient of powers property?

A: The quotient of powers property is a mathematical property that states that when you divide two exponential expressions with the same base, you can subtract their exponents. For example, if you have the equation $\frac{a^m}{a^n} = a^{m-n}$ , then the quotient of powers property allows you to simplify the equation by subtracting the exponents.

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 can use the properties of exponents to simplify the equation and then equate the exponents to solve for the variable. For example, if you have the equation $a^x \cdot a^y = a^z$ , then you can use the product of powers property to simplify the equation and then equate the exponents to solve for x.

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

A: An exponential equation is an equation that involves an exponential expression, while a logarithmic equation is an equation that involves a logarithmic expression. Logarithmic expressions are the inverse of exponential expressions, and they are used to solve equations that involve exponential expressions.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to simplify the equation and then equate the logarithmic expressions to solve for the variable. For example, if you have the equation $\log_a x = y$ , then you can use the properties of logarithms to simplify the equation and then equate the logarithmic expressions to solve for x.

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 the product of powers property to simplify the equation
Not using the quotient of powers property to simplify the equation
Not equating the exponents to solve for the variable
Not checking the domain of the equation to ensure that the solution is valid

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises in a textbook or online resource. You can also try solving exponential equations on your own by creating your own examples and exercises. Additionally, you can use online resources such as Khan Academy or Mathway to practice solving exponential equations.

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

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

Modeling population growth and decay
Modeling financial growth and decay
Modeling chemical reactions and decay
Modeling electrical circuits and decay

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

A: To use exponential equations to model real-world problems, you can start by identifying the variables and parameters involved in the problem. You can then use the properties of exponents to simplify the equation and solve for the variable. Finally, you can use the solution to the equation to make predictions or recommendations about the real-world problem.