Question

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[4]{\frac{7}{8}}$$

Short answer

\(\frac{\sqrt[4]{7}}{2^{\frac{3}{4}}}\)

Step-by-step solution

Understand the Problem

We are required to write the given fourth root in simplest radical form without using a calculator. The given expression is \(\sqrt[4]{\frac{7}{8}}\).

Break Down the Numerator and Denominator

First, separate the fourth root of the numerator and the denominator into distinct fourth roots: \(\sqrt[4]{\frac{7}{8}} = \frac{\sqrt[4]{7}}{\sqrt[4]{8}}\).

Simplify the Fourth Root of the Denominator

Recognize that 8 is a perfect fourth power since \(8 = 2^3\) and can be written as \(2^3 = (2^2)^\frac{3}{2}\). Therefore, \(\sqrt[4]{8} = \sqrt[4]{2^3} = 2^{\frac{3}{4}}\). Since the exponent \(\frac{3}{4}\) is less than 1, the expression cannot be simplified further.

Combine the Results

The numerator \(\sqrt[4]{7}\) cannot be further simplified as 7 is not a perfect fourth. Therefore, the simplest form of the original expression remains as \(\frac{\sqrt[4]{7}}{2^{\frac{3}{4}}}\) or can be written as \(\frac{\sqrt[4]{7}}{\sqrt[4]{8}}\) as would be typical in radical form.

Key concepts

Fourth Root

The fourth root of a number, denoted as \(\sqrt[4]{x}\), is a value that, when raised to the fourth power, gives the original number x. In other words, if \(a = \sqrt[4]{x}\), then \(a^4 = x\). Fourth roots are a specific type of root, similar to the more commonly known square root (second root).

Understanding fourth roots is key when working with radical expressions, especially when the problem involves variables raised to the fourth power or when simplifying fractions that contain fourth roots. Unlike square roots, which have one positive and one negative root, fourth roots have two positive and two negative roots, because the process of raising to the fourth power is even (neglecting complex roots in this context). However, in most algebraic contexts, we're only interested in the principal (positive) root, particularly when dealing with real numbers.

Radical Form

Expressing numbers in radical form means writing them with roots, such as square roots, cube roots, fourth roots, and so on, instead of using fractional exponents. The radical symbol \(\sqrt{ }\) is used to denote roots, with an index to indicate the type of root: \(\sqrt[2]{ }\) for square roots, \(\sqrt[3]{ }\) for cube roots, and \(\sqrt[4]{ }\) for fourth roots, for instance.

Keeping expressions in radical form can be helpful for visually understanding the properties of the number or expression inside the root sign. When simplifying these expressions, the goal is to maintain the form while reducing the expression to its simplest terms. In algebra, it's common to work with expressions in radical form, particularly when precision is required, as rounding can introduce errors compared to exact radical values.

Perfect Powers

Perfect powers refer to numbers that can be expressed as an integer raised to the power of another integer. For example, \(8 = 2^3\) is a perfect power because it is 2 raised to the 3rd power. This becomes particularly useful when simplifying radicals, as recognizing perfect powers allows for the radical to be simplified to an integer or a simpler radical.

For instance, in the context of fourth roots, if a number is a perfect fourth power—like \(16 = 2^4\) or \(81 = 3^4\)—its fourth root will simplify to an integer (2 or 3 respectively). When dealing with radicals, always look for the highest perfect power that divides into your number to simplify the radical as much as possible.

Simplify Radicals Without Calculator

To simplify radicals without a calculator involves several techniques, including factoring out perfect powers and rationalizing denominators. Understanding the fundamentals of radicals and their properties is essential to execute this process. The technique usually includes breaking down the number inside the radical to its prime factorization and then grouping the factors based on the type of root.

For simplifying fourth roots, you would look for groups of four identical factors, moving one of each group outside the radical. Any remaining factors that aren't part of a complete group stay inside the radical. This visual method is often more intuitive than working with fractional exponents, and it's quite manageable without a calculator once you're familiar with the properties of radicals and perfect powers.