Simplifying Expressions With Negative Exponents Rewriting And Simplifying $81^{-\frac{1}{4}}$

Understanding Negative Exponents

When dealing with exponents, a negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words, for any non-zero number 'a' and any integer 'n', we have the following relationship:

$a^{-n} = \frac{1}{a^n}$

This property is crucial for rewriting expressions with negative exponents into expressions with positive exponents. Understanding this principle allows us to manipulate and simplify mathematical expressions more effectively. It's a foundational concept in algebra and is essential for solving various mathematical problems. The ability to convert negative exponents to positive exponents not only simplifies calculations but also provides a clearer understanding of the underlying mathematical relationships. This concept is widely used in various fields, including physics, engineering, and computer science, where exponential functions and their properties are frequently encountered. By mastering this principle, one can confidently tackle more complex mathematical challenges involving exponents and radicals.

Rewriting $81^{-\frac{1}{4}}$ with a Positive Exponent

To rewrite the expression $81^{-\frac{1}{4}}$ with a positive exponent, we apply the rule mentioned above. Here, our base is 81, and our exponent is $-\frac{1}{4}$. Using the property $a^{-n} = \frac{1}{a^n}$, we can rewrite the expression as follows:

$81^{-\frac{1}{4}} = \frac{1}{81^{\frac{1}{4}}}$

Now, we have successfully rewritten the expression with a positive exponent, which is $\frac{1}{4}$. This transformation is the first step in simplifying the expression. The positive exponent makes it easier to understand the operation we need to perform, which in this case involves finding a root of the base. The transformation from a negative exponent to a positive exponent is a fundamental technique in simplifying mathematical expressions and is widely used in various mathematical contexts. By applying this rule, we can convert complex expressions into simpler forms that are easier to evaluate and manipulate. This skill is essential for anyone working with exponents and radicals, as it allows for a more intuitive understanding of the mathematical relationships involved.

Simplifying the Expression

Now that we have $81^{-\frac{1}{4}} = \frac{1}{81^{\frac{1}{4}}}$, we need to simplify $81^{\frac{1}{4}}$. An exponent of $\frac{1}{4}$ indicates the fourth root of the base. In other words, we are looking for a number that, when raised to the fourth power, equals 81.

Mathematically, this can be written as:

$81^{\frac{1}{4}} = \sqrt[4]{81}$

We need to find a number 'x' such that $x^4 = 81$. Let's consider the factors of 81. We know that 81 can be expressed as $3 \times 3 \times 3 \times 3$, which is $3^4$. Therefore:

$81^{\frac{1}{4}} = \sqrt[4]{3^4} = 3$

Now, we substitute this back into our expression:

$\frac{1}{81^{\frac{1}{4}}} = \frac{1}{3}$

Thus, the simplified form of $81^{-\frac{1}{4}}$ is $\frac{1}{3}$. This simplification process demonstrates the power of understanding fractional exponents and their relationship to radicals. By breaking down the problem into smaller steps, we can systematically simplify complex expressions. The ability to recognize perfect powers and roots is crucial in this process. In this case, recognizing that 81 is a perfect fourth power of 3 allowed us to quickly simplify the expression. This skill is essential in various mathematical contexts, including algebra, calculus, and number theory. Furthermore, understanding how to simplify expressions with fractional exponents is essential for solving equations and modeling real-world phenomena.

Final Answer

Therefore, $81^{-\frac{1}{4}}$ rewritten with a positive exponent and simplified is $\frac{1}{3}$. This result showcases the importance of understanding the properties of exponents and radicals in simplifying mathematical expressions. The process involves converting the negative exponent to a positive exponent, recognizing the fractional exponent as a root, and then simplifying the radical. This method can be applied to various similar problems, making it a valuable tool in mathematical problem-solving. The final answer not only provides a numerical result but also demonstrates the application of mathematical principles and techniques. By understanding these principles, one can confidently tackle more complex mathematical challenges involving exponents and radicals.

In summary, we have successfully rewritten $81^{-\frac{1}{4}}$ with a positive exponent as $\frac{1}{81^{\frac{1}{4}}}$ and then simplified it to $\frac{1}{3}$. This process involved understanding the relationship between negative exponents, positive exponents, and radicals. The ability to manipulate exponents and radicals is a fundamental skill in mathematics and is essential for solving a wide range of problems. This example illustrates how a seemingly complex expression can be simplified by applying basic mathematical principles. The final result provides a clear and concise answer to the problem, demonstrating the power of mathematical simplification. This skill is not only valuable in academic settings but also in various real-world applications where mathematical models and calculations are used. By mastering these concepts, one can confidently approach mathematical challenges and solve them effectively.