If $14^x=\sqrt[13]{14}$, What Value Of $x$ Makes This Equation True?A. $\frac{13}{14}$ B. $ 1 13 \sqrt[13]{1} 13 1 [/tex] C. $13$ D. $\frac{1}{13}$

Mar 12, 2025 by ADMIN 163 views

Solving the Equation: Uncovering the Value of x

In this article, we will delve into the world of mathematics and explore a fascinating equation involving exponents and roots. The equation in question is $14^x=\sqrt[13]{14}$, and our goal is to determine the value of $x$ that makes this equation true. We will employ various mathematical techniques to solve for $x$ and arrive at the correct answer.

Before we begin solving the equation, let's take a closer look at its components. The left-hand side of the equation is $14^x$, which represents an exponential expression with base $14$ and exponent $x$. The right-hand side is $\sqrt[13]{14}$, which is a radical expression with index $13$ and radicand $14$.

To simplify the equation and make it easier to solve, we can rewrite the right-hand side using exponent notation. Recall that $\sqrt[n]{a} = a^{\frac{1}{n}}$, so we can rewrite $\sqrt[13]{14}$ as $14^{\frac{1}{13}}$.

14^x = 14^{\frac{1}{13}}

Now that we have rewritten the equation, we can equate the exponents on both sides. Since the bases are the same (both are $14$), we can set the exponents equal to each other:

x = \frac{1}{13}

To ensure that our solution is correct, let's plug it back into the original equation and verify that it satisfies the equation. Substituting $x = \frac{1}{13}$ into the original equation, we get:

14^{\frac{1}{13}} = \sqrt[13]{14}

This equation is indeed true, as we can verify by raising $14$ to the power of $\frac{1}{13}$ and obtaining $\sqrt[13]{14}$.

In this article, we have solved the equation $14^x=\sqrt[13]{14}$ and determined the value of $x$ that makes this equation true. By rewriting the equation using exponent notation and equating the exponents, we arrived at the solution $x = \frac{1}{13}$. We have also verified that this solution satisfies the original equation, providing confidence in our answer.

The correct answer is:

D. $\frac{1}{13}$

When solving equations involving exponents and roots, it's essential to rewrite the equation using exponent notation to simplify the problem.
Equating exponents is a powerful technique for solving equations involving the same base.
Verifying the solution by plugging it back into the original equation is a crucial step in ensuring that the solution is correct.

Solving equations involving exponents and roots requires a deep understanding of mathematical concepts and techniques. By employing the strategies outlined in this article, you can develop the skills and confidence needed to tackle complex mathematical problems. Remember to always verify your solutions and to be patient and persistent in your problem-solving endeavors.
Frequently Asked Questions: Solving the Equation $14^x=\sqrt[13]{14}$

In our previous article, we explored the equation $14^x=\sqrt[13]{14}$ and determined the value of $x$ that makes this equation true. In this article, we will address some of the most frequently asked questions related to this equation and provide additional insights and explanations.

A: The base 14 in the equation is a crucial component that affects the value of $x$. The base 14 represents the number that is being raised to the power of $x$. In this case, the base 14 is being raised to the power of $x$ to produce the value $\sqrt[13]{14}$.

A: The exponent $x$ is essential in the equation because it determines the power to which the base 14 is raised. The value of $x$ affects the resulting value of the equation, and in this case, it is used to produce the value $\sqrt[13]{14}$.

A: The index 13 in the radical $\sqrt[13]{14}$ affects the equation by determining the power to which the base 14 is raised. In this case, the index 13 is used to produce the value $\sqrt[13]{14}$, which is equivalent to $14^{\frac{1}{13}}$.

A: The exponent $x$ and the index 13 are related in that they both affect the power to which the base 14 is raised. In this case, the exponent $x$ is equal to $\frac{1}{13}$, which is the reciprocal of the index 13.

A: To verify that the solution $x = \frac{1}{13}$ is correct, you can plug it back into the original equation and simplify. If the resulting value is equal to $\sqrt[13]{14}$, then the solution is correct.

A: Some common mistakes to avoid when solving equations involving exponents and roots include:

Failing to rewrite the equation using exponent notation
Not equating the exponents correctly
Not verifying the solution by plugging it back into the original equation

A: To apply the techniques used in this article to other problems involving exponents and roots, you can follow these steps:

Rewrite the equation using exponent notation
Equate the exponents correctly
Verify the solution by plugging it back into the original equation

In this article, we have addressed some of the most frequently asked questions related to the equation $14^x=\sqrt[13]{14}$ and provided additional insights and explanations. By following the techniques outlined in this article, you can develop the skills and confidence needed to tackle complex mathematical problems involving exponents and roots.

When solving equations involving exponents and roots, it's essential to rewrite the equation using exponent notation to simplify the problem.
Equating exponents is a powerful technique for solving equations involving the same base.
Verifying the solution by plugging it back into the original equation is a crucial step in ensuring that the solution is correct.