Problem 32 - Algebra 1

Question

Simplify the expression. $$5 \sqrt{\frac{6}{2}}$$

Step-by-Step Solution

Answer

The simplification of the given expression is \(5 \sqrt{3}\).

Step 1: Perform the division

We can see that the fraction inside the square root is \(\frac{6}{2}\). So, we divide the numerator by the denominator to get \(3\). This results in the following simplification: \(5 \sqrt{3}\)

Step 2: Solve the square root

We can now proceed to simplify the square root. Since square root of 3 can't be simplified further, we can safely assume that our simplified mathematical expression is \(5 \sqrt{3}\).

Key Concepts

Square Roots

A square root is an essential mathematical operation often used when dealing with expressions that require simplification. It's all about finding a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. In more general terms, the square root of a number 'x' is represented as \(\sqrt{x}\), indicating the number that, when squared, will yield 'x'. Some numbers, like 4 or 9, have perfect square roots (2 and 3, respectively). However, many numbers, such as 3 in our case, do not yield whole numbers when their square roots are calculated.These non-perfect square roots often remain in their radical form, as they can't be neatly expressed as integers or simple fractions. For instance, since \(\sqrt{3}\) is an irrational number, it can't be simplified further and remains as \(\sqrt{3}\) in expressions.

Division

Division is one of the basic arithmetic operations used to split a number into equal parts. It involves distributing a given number, known as the dividend, by another number, the divisor, to achieve a quotient. In our exercise, division plays a vital role in simplifying the expression inside the square root.The original expression \(\frac{6}{2}\) appears under the square root, representing the quantity of 6 divided by 2. By performing this division, we find that 6 divided by 2 equals 3, resulting in a simpler expression \(\sqrt{3}\) inside the square root.This simplification step is crucial because it reduces the complexity of the expression and allows for easier handling of the square root operation. Without performing division, the task of simplifying radicals would be much more cumbersome.

Simplifying Radicals

Simplifying radicals is the process of breaking down expressions involving roots into their most basic and easy-to-understand forms. In expressions with square roots, like \(5 \sqrt{\frac{6}{2}}\), the goal is to express the radical in a simpler or more readable format.By simplifying the fraction inside the radical, as demonstrated with \(\frac{6}{2}\) to get 3, we move closer to simplifying the radical expression itself. Since \(\sqrt{3}\) can't be broken down further due to it being a non-perfect square, our expression \(5 \sqrt{3}\) is now fully simplified.Simplifying radicals ensures that expressions are written elegantly, allowing both easier calculation and clearer communication of mathematical ideas. It involves these steps:

• Perform any possible arithmetic inside the radical first, like division.
• Simplify the square root, if possible, leaving it in the radical form if it cannot be reduced further.
• Combine the simplified radical with any coefficients outside the radical to complete the simplification.

These steps help in condensing expressions, making them easier to work with in further mathematical operations.