Finding The Root Of The Fifth Root Of 1/32 A Step By Step Guide

Embark on a mathematical journey to uncover the root of the intriguing expression $\sqrt[5]{\frac{1}{32}}$. This exploration delves into the fundamental concepts of radicals, exponents, and fractions, providing a step-by-step guide to arrive at the solution. Understanding these core principles is crucial for mastering various mathematical operations and problem-solving techniques.

Dissecting the Expression: Radicals, Exponents, and Fractions

To effectively tackle the problem, let's first dissect the expression and understand its components. The expression $\sqrt[5]{\frac{1}{32}}$ involves a fifth root, a fraction, and the interplay between radicals and exponents. A radical, denoted by the symbol $\sqrt[n]{}$, represents the inverse operation of exponentiation. In this case, we have a fifth root, meaning we are looking for a number that, when raised to the power of 5, equals the radicand (the expression under the radical symbol), which is $\frac{1}{32}$.

A fraction, on the other hand, represents a part of a whole. The fraction $\frac{1}{32}$ indicates one part out of 32 equal parts. Understanding fractions is essential for performing arithmetic operations and solving equations. The connection between radicals and exponents is a cornerstone of mathematics. The nth root of a number can be expressed as a fractional exponent. Specifically, $\sqrt[n]{x}$ is equivalent to $x^{\frac{1}{n}}$. This equivalence allows us to manipulate radical expressions using the rules of exponents.

Prime Factorization: Unraveling the Radicand

The next step in solving the problem is to simplify the radicand, $\frac{1}{32}$. To do this, we can employ the powerful technique of prime factorization. Prime factorization involves breaking down a number into its prime factors, which are prime numbers that, when multiplied together, give the original number. In the case of 32, its prime factorization is $2 \times 2 \times 2 \times 2 \times 2$, which can be written more compactly as $2^5$. Therefore, the fraction $\frac{1}{32}$ can be expressed as $\frac{1}{2^5}$. This representation is crucial because it allows us to leverage the relationship between exponents and radicals.

By expressing the denominator as a power of 2, we can rewrite the original expression as $\sqrt[5]{\frac{1}{2^5}}$. Now, we can use the property of exponents that states $\frac{1}{a^n} = a^{-n}$. Applying this property, we get $\frac{1}{2^5} = 2^{-5}$. Substituting this back into the expression, we have $\sqrt[5]{2^{-5}}$. This transformation is a key step towards simplifying the expression and finding its root.

The Interplay of Radicals and Exponents: Simplifying the Expression

With the radicand expressed as a power with a negative exponent, we can now utilize the fundamental connection between radicals and exponents. Recall that $\sqrt[n]{x}$ is equivalent to $x^{\frac{1}{n}}$. Applying this equivalence to our expression, $\sqrt[5]{2^{-5}}$, we can rewrite it as $(2^{-5}){\frac{1}{5}}$. This transformation is a crucial step in simplifying the expression and isolating the root.

Now, we can employ another important property of exponents: $(a^m)n = a^{m \times n}$. This property states that when raising a power to another power, we multiply the exponents. Applying this property to our expression, $(2^{-5}){\frac{1}{5}}$, we multiply the exponents -5 and $\frac{1}{5}$, which gives us $-5 \times \frac{1}{5} = -1$. Therefore, our expression simplifies to $2^{-1}$. This simplification is a significant step towards finding the final answer.

Unveiling the Root: The Final Calculation

We have successfully simplified the original expression to $2^{-1}$. To find the root, we need to evaluate this expression. Recall that a negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. Applying this property to $2^{-1}$, we get $2^{-1} = \frac{1}{2^1}$. Since $2^1 = 2$, the expression further simplifies to $\frac{1}{2}$. Therefore, the root of $\sqrt[5]{\frac{1}{32}}$ is $\frac{1}{2}$.

This final calculation reveals the solution to our initial problem. By systematically applying the principles of radicals, exponents, and fractions, we have successfully determined the root of the given expression. This process highlights the interconnectedness of these mathematical concepts and the importance of mastering them for problem-solving.

Conclusion: The Power of Mathematical Principles

In conclusion, the root of $\sqrt[5]{\frac{1}{32}}$ is $\frac{1}{2}$. This solution was obtained by meticulously applying the principles of radicals, exponents, and fractions. The process involved prime factorization, conversion between radicals and fractional exponents, and the application of exponent rules. This exploration underscores the power of mathematical principles in simplifying complex expressions and arriving at accurate solutions. By understanding and applying these core concepts, we can confidently tackle a wide range of mathematical challenges.

Answer

(B) $rac{1}{2}$