Solving for c and d in the Exponential Equation

Oct 13, 2025 by magicalceylontours.com 17 views

Solving for c and d in the Exponential Equation

In this article, we will delve into solving for the values of $c$ and $d$ in the given exponential equation. The equation involves variables $x$ and $y$ , and we need to find the values of $c$ and $d$ that make the equation true. We will use algebraic manipulation and properties of exponents to solve for $c$ and $d$ .

The given equation is:

$\left(\frac{x}{y^d}\right)^c\left(\frac{x^2}{y^5}\right)=x^7$

Our goal is to find the values of $c$ and $d$ that satisfy this equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by combining the two fractions on the left-hand side:

$\left(\frac{x}{y^d}\right)^c\left(\frac{x^2}{y^5}\right) = \frac{x^{c+2}}{y^{5+cd}}$

Now, the equation becomes:

$\frac{x^{c+2}}{y^{5+cd}} = x^7$

Step 2: Equate the Exponents

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

$c+2 = 7$

and

$5+cd = 0$

Step 3: Solve for c

From the first equation, we can solve for $c$ :

$c = 7 - 2$

$c = 5$

Step 4: Solve for d

Now that we have the value of $c$ , we can substitute it into the second equation:

$5+5d = 0$

Subtracting 5 from both sides gives:

$5d = -5$

Dividing both sides by 5 gives:

$d = -1$

In conclusion, the values of $c$ and $d$ that make the equation true are:

$c = 5$

$d = -1$

Therefore, the solution to the equation is $c=5,d=-1$ .
Frequently Asked Questions (FAQs) about Solving for c and d in the Exponential Equation

In our previous article, we solved for the values of $c$ and $d$ in the given exponential equation. However, we understand that some readers may still have questions about the solution. In this article, we will address some of the frequently asked questions (FAQs) about solving for $c$ and $d$ in the exponential equation.

Q: What is the significance of the values of c and d in the exponential equation?

A: The values of $c$ and $d$ in the exponential equation are crucial in determining the behavior of the function. In this case, we found that $c=5$ and $d=-1$ . These values indicate that the function will grow rapidly as $x$ increases, but will eventually decrease as $y$ increases.

Q: How did you simplify the equation in Step 1?

A: We simplified the equation by combining the two fractions on the left-hand side. This involved multiplying the numerators and denominators of the fractions together.

Q: Why did you equate the exponents in Step 2?

A: We equated the exponents because the bases of the exponents are the same. This is a fundamental property of exponents, which states that if two exponential expressions have the same base, then their exponents must be equal.

Q: How did you solve for c in Step 3?

A: We solved for $c$ by subtracting 2 from both sides of the equation $c+2=7$ . This gave us the value of $c$ as $c=5$ .

Q: How did you solve for d in Step 4?

A: We solved for $d$ by substituting the value of $c$ into the equation $5+5d=0$ . This gave us the value of $d$ as $d=-1$ .

Q: What if the equation had been more complex? How would you have solved for c and d?

A: If the equation had been more complex, we would have used a variety of techniques to simplify it, such as factoring, combining like terms, and using algebraic manipulations. We would have also used properties of exponents and logarithms to solve for $c$ and $d$ .

Q: Can you provide more examples of exponential equations with multiple variables?

A: Yes, here are a few examples of exponential equations with multiple variables:

$\left(\frac{x}{y^d}\right)^c\left(\frac{x^2}{y^5}\right)=x^7$
$\left(\frac{x}{y^d}\right)^c\left(\frac{x^2}{y^5}\right)=y^3$
$\left(\frac{x}{y^d}\right)^c\left(\frac{x^2}{y^5}\right)=x^2y^2$

These equations can be solved using similar techniques to the one we used in this article.

In conclusion, we hope that this article has addressed some of the frequently asked questions about solving for $c$ and $d$ in the exponential equation. If you have any further questions or need additional clarification, please don't hesitate to ask.