Solve The Equation:$\[ \frac{(7x)^2}{7^5} = 7^7 \\]

Introduction

In this article, we will delve into the world of mathematics and explore a fascinating equation that involves exponents and fractions. The equation in question is $\frac{(7x)^2}{7^5} = 7^7$. Our goal is to solve for the variable $x$ and understand the underlying mathematical concepts that make this equation tick.

Understanding Exponents

Before we dive into the solution, let's take a moment to review the concept of exponents. An exponent is a small number that is raised to a power, indicating how many times a base number should be multiplied by itself. For example, $7^3$ means $7$ multiplied by itself three times: $7 \times 7 \times 7 = 343$. Exponents are a fundamental concept in mathematics, and they play a crucial role in solving equations like the one we're dealing with.

Breaking Down the Equation

Now that we have a solid understanding of exponents, let's break down the equation and identify the key components. The equation is $\frac{(7x)^2}{7^5} = 7^7$. We can see that the left-hand side of the equation involves a fraction, with the numerator being $(7x)^2$ and the denominator being $7^5$. The right-hand side of the equation is simply $7^7$.

Step 1: Simplify the Left-Hand Side

To simplify the left-hand side of the equation, we can start by expanding the numerator using the exponent rule $(ab)^n = a^n \cdot b^n$. Applying this rule, we get:

$$\frac{(7x)^2}{7^5} = \frac{7^2 \cdot x^2}{7^5}$$

Next, we can simplify the fraction by canceling out common factors in the numerator and denominator. In this case, we can cancel out $7^2$ from both the numerator and denominator:

$$\frac{7^2 \cdot x^2}{7^5} = \frac{x^2}{7^3}$$

Step 2: Equate the Left-Hand Side to the Right-Hand Side

Now that we have simplified the left-hand side of the equation, we can equate it to the right-hand side:

$$\frac{x^2}{7^3} = 7^7$$

Step 3: Eliminate the Fraction

To eliminate the fraction, we can multiply both sides of the equation by $7^3$:

$$x^2 = 7^7 \cdot 7^3$$

Step 4: Simplify the Right-Hand Side

Using the exponent rule $a^m \cdot a^n = a^{m+n}$, we can simplify the right-hand side of the equation:

$$x^2 = 7^{7+3}$$

$$x^2 = 7^{10}$$

Step 5: Take the Square Root

To solve for $x$, we can take the square root of both sides of the equation:

$$x = \sqrt{7^{10}}$$

Step 6: Simplify the Square Root

Using the exponent rule $\sqrt{a^n} = a^{n/2}$, we can simplify the square root:

$$x = 7^{10/2}$$

$$x = 7^5$$

Conclusion

And there you have it! We have successfully solved the equation $\frac{(7x)^2}{7^5} = 7^7$ and found that $x = 7^5$. This equation may seem daunting at first, but by breaking it down into smaller steps and using exponent rules, we were able to simplify the equation and arrive at a solution.

Real-World Applications

While this equation may seem like a purely theoretical exercise, it has real-world applications in fields such as physics and engineering. For example, in the study of electrical circuits, exponents are used to describe the behavior of resistors and capacitors. By understanding how to solve equations like this one, we can gain a deeper understanding of the underlying mathematical concepts that govern the behavior of these systems.

Final Thoughts

Q: What is the main concept behind solving the equation $\frac{(7x)^2}{7^5} = 7^7$?

A: The main concept behind solving this equation is understanding exponents and how to simplify fractions using exponent rules.

Q: What is the first step in simplifying the left-hand side of the equation?

A: The first step in simplifying the left-hand side of the equation is to expand the numerator using the exponent rule $(ab)^n = a^n \cdot b^n$. This gives us $\frac{7^2 \cdot x^2}{7^5}$.

Q: How do we simplify the fraction on the left-hand side of the equation?

A: We simplify the fraction by canceling out common factors in the numerator and denominator. In this case, we can cancel out $7^2$ from both the numerator and denominator, resulting in $\frac{x^2}{7^3}$.

Q: What is the next step in solving the equation?

A: The next step is to equate the left-hand side of the equation to the right-hand side: $\frac{x^2}{7^3} = 7^7$.

Q: How do we eliminate the fraction on the left-hand side of the equation?

A: We eliminate the fraction by multiplying both sides of the equation by $7^3$, resulting in $x^2 = 7^7 \cdot 7^3$.

Q: What is the final step in solving the equation?

A: The final step is to take the square root of both sides of the equation, resulting in $x = \sqrt{7^{10}}$. Simplifying the square root gives us $x = 7^5$.

Q: What are some real-world applications of solving equations like this one?

A: Solving equations like this one has real-world applications in fields such as physics and engineering. For example, in the study of electrical circuits, exponents are used to describe the behavior of resistors and capacitors.

Q: What are some tips for solving equations like this one?

A: Some tips for solving equations like this one include:

• Breaking down the equation into smaller steps
• Using exponent rules to simplify fractions
• Canceling out common factors in the numerator and denominator
• Taking the square root of both sides of the equation

Q: What is the most important thing to remember when solving equations like this one?

A: The most important thing to remember when solving equations like this one is to be patient and take your time. Solving equations like this one requires a combination of mathematical knowledge, problem-solving skills, and patience.

Q: Can you provide more examples of equations like this one?

A: Yes, here are a few more examples of equations like this one:

• $\frac{(3x)^2}{3^5} = 3^7$
• $\frac{(2x)^2}{2^5} = 2^7$
• $\frac{(5x)^2}{5^5} = 5^7$

These equations can be solved using the same steps as the original equation.