Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents. $$ 81^{-1 / 4} $$

Step-by-Step Solution

Answer

The short answer based on the given step-by-step solution is: \(81^{-\frac{1}{4}} = \frac{1}{3}\).

Step 1: Simplify the exponent

We are given a fractional exponent \(-1/4\), which can be expressed as a combination of an exponent and a root: \[81^{-\frac{1}{4}} = \sqrt[4]{81^{-1}}\]. Next, we will find the fourth root of 81 and then take its reciprocal.

Step 2: Find the fourth root of 81

To find the fourth root of 81, we look for a number, that when multiplied by itself four times, results in 81. In this case, that number is 3: \[\sqrt[4]{81} = 3^{4} = 3\]

Step 3: Take the reciprocal of the fourth root of 81

We can now find the reciprocal of the fourth root of 81 by finding the reciprocal of the result we found in step 2: \[81^{-\frac{1}{4}} = (\sqrt[4]{81})^{-1} = 3^{-1}\]

Step 4: Simplify the resulting expression

Finally, the last step is to simplify expression \(3^{-1}\). The negative sign in the exponent indicates finding the reciprocal of 3: \[3^{-1} = \frac{1}{3}\] So, our final answer is: \[81^{-\frac{1}{4}} = \frac{1}{3}\].

Key Concepts

Fractional Exponents

A fractional exponent is a way of expressing powers and roots together in one compact form. It's like a shortcut to show both operations happening at once. For example, if you have the expression \(a^{m/n}\), it means you're doing two things:

• First, you raise \(a\) to the power of \(m\).
• Then, you take the \(n\)th root of the result.

The numerator in the fraction (\(m\)) represents the power. The denominator (\(n\)) represents the root. So, \(81^{-1/4}\) means you take the fourth root of 81, and then because of the negative sign, you find the reciprocal of that result.
Understanding the role of fractional exponents provides insight into how complex expressions can be unraveled. This is a fundamental skill in algebra and higher-level mathematics.

Reciprocal

The reciprocal of a number is one divided by that number. It flips the number over. For example, the reciprocal of 3 is \(\frac{1}{3}\). To find the reciprocal of a fraction, you simply swap the numerator and the denominator. So, the reciprocal of \(\frac{2}{3}\) would be \(\frac{3}{2}\).

In the context of the expression \(81^{-1/4}\), the negative exponent tells us to take the reciprocal after finding the fourth root of 81. Exponents with negative signs are a way to convey the concept of reciprocals in mathematical expressions. For \(3^{-1}\), it directly translates to the reciprocal \(\frac{1}{3}\).
Understanding reciprocals is crucial when handling negative exponents in math. Reciprocals are part of simplifying radical expressions and solving equations.

Fourth Root

A fourth root asks for the number which, when multiplied by itself four times, equals the original number. It's similar to square roots, but instead of multiplying by itself twice, you do it four times. To express the fourth root mathematically, we use \(\sqrt[4]{x}\) or \(x^{1/4}\).
For example, the fourth root of 81 is 3 because \(3 \times 3 \times 3 \times 3 = 81\). In step-by-step math problems, the fourth root plays a pivotal role in transformations using fractional exponents. It's essential to understand roots as part of solving problems like \(81^{-1/4}\).
Recognizing the various roots helps you break down and simplify different mathematical expressions, making them manageable and easier to understand.