Question

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{\frac{5 m}{7 n}}$$

Short answer

\(\sqrt{\frac{5m}{7n}}\) cannot be simplified further and is in its simplest form.

Step-by-step solution

Simplify the Radical Expression

To simplify the given radical expression, \(\sqrt{\frac{5m}{7n}}\), you should look for perfect squares that can be taken out of the square root. However, since 5 and 7 are prime numbers and there are no variables with exponents that are multiples of 2, you cannot simplify the expression by taking anything outside of the square root.

Express Separately

You can express the radical as a product of two separate radicals for the numerator and the denominator to make it easier to identify if there's any simplification possible: \(\sqrt{\frac{5m}{7n}} = \frac{\sqrt{5m}}{\sqrt{7n}}\).

Examine Numerator and Denominator for Simplification

The radical in the numerator \(\sqrt{5m}\) and the radical in the denominator \(\sqrt{7n}\) are both in simplest form as there are no perfect square factors for 5, 7, m, or n assuming that m and n are not squares or multiples of perfect squares. Thus, the expression is already in its simplest form.

Key concepts

Square Root Simplification

Square root simplification involves reducing a radical expression into its simplest form without changing its value. The process generally requires identifying and extracting perfect square factors from under the radical sign. A perfect square is a number or expression that is the square of an integer, like 4, 9, or 16, which are the squares of 2, 3, and 4 respectively.

When simplifying the square root of a fraction, like \(\sqrt{\frac{5m}{7n}}\), the goal is to look inside both the numerator and the denominator for any perfect squares. However, as 5 and 7 are prime numbers and the variables 'm' and 'n' are not raised to any power that suggests they are perfect squares, the expression cannot be simplified further. The lack of perfect squares indicates that the radical expression is already in its simplest possible form. It's important to note that we must keep the expression in radical form without resorting to decimal approximations.

Radical Form

The radical form is a way of expressing numbers or expressions that include roots, like square roots or cube roots. It is often preferable in mathematics to maintain the integrity of the root, providing an exact value rather than an approximation.

In the expression \(\sqrt{\frac{5m}{7n}}\), breaking it down into \(\frac{\sqrt{5m}}{\sqrt{7n}}\) denotes separating the radical over the numerator and the denominator. This structure allows for clearer inspection of each part for potential simplification and for applying the properties of radicals. While the extraction of squares wasn't possible, this step shows how the expression can be worked on while keeping the values within the radicals intact. By expressing radicals in this way, we adhere to the principles of presenting exact values and promoting clearer understanding.

Mathematical Radicals

Mathematical radicals are symbols used to indicate the root of a number or expression. The most common radical is the square root, but radicals can also represent higher order roots, such as cube roots or fourth roots. It's essential to understand that a radical can sometimes be simplified (when containing perfect squares) or left as is when no further simplification is possible.

In our exercise, \(\sqrt{\frac{5m}{7n}}\) is a square root (2nd root) symbol indicating the radical involving variables. Since no simplification of individual elements within the radical was possible, the expression itself embodies the fundamental characteristics of radicals in mathematics: to represent the root of quantities in a compact and exact form. The approach to simplifying radicals like this aligns with the core principle in math that expressions should be as simple as possible but no simpler than the math allows.