Problem 5 - Technical Mathematics with Calculus

Question

Addition and Subtraction of Radicals. Combine as indicated and simplify. $$2 \sqrt{50}+\sqrt{72}+3 \sqrt{18}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(25\sqrt{2}\).

Step 1 - Factor Inside Radicals

Break down the numbers inside the square roots into their prime factors: \(50 = 2 \times 5^2\), \(72 = 2^3 \times 3^2\), and \(18 = 2 \times 3^2\).

Step 2 - Simplify Radicals

Take out the squared factors outside of the square root as single numbers: \(2 \sqrt{50} = 2 \sqrt{2 \times 5^2} = 2 \times 5 \sqrt{2} = 10\sqrt{2}\), \(\sqrt{72} = \sqrt{2^3 \times 3^2} = 2 \times 3 \sqrt{2} = 6\sqrt{2}\), and \(3 \sqrt{18} = 3 \sqrt{2 \times 3^2} = 3 \times 3 \sqrt{2} = 9\sqrt{2}\).

Step 3 - Combine Like Radicals

Add the coefficients of like radicals together: \(10\sqrt{2} + 6\sqrt{2} + 9\sqrt{2} = (10 + 6 + 9)\sqrt{2} = 25\sqrt{2}\).

Key Concepts

Addition and Subtraction of Radicals

When we talk about adding and subtracting radicals, it's essential to understand that just like with algebraic terms, only like radicals can be combined. A radical is considered 'like' with another if they have the same index and the same radicand (the number or expression underneath the radical sign).
To add or subtract radicals, follow these general steps: first, simplify the radicals involved if possible, then combine the like radicals.

Example:

In the case of our exercise, once we've simplified the radicals, we have terms like \(10\sqrt{2}\), \(6\sqrt{2}\), and \(9\sqrt{2}\) which can be added together because they all contain the radical \(\sqrt{2}\). The numbers in front of the radical are known as coefficients, and these can be added just like ordinary numbers, resulting in \(25\sqrt{2}\). Remember, if the radicals were not alike, such as \(\sqrt{2}\) and \(\sqrt{3}\), they could not be combined through addition or subtraction.

Factoring Numbers

Factoring is a foundational skill in mathematics, used to break down numbers into their constituent factors. Factors are whole numbers that can be multiplied together to produce another number.
When simplifying expressions, especially radicals, it’s beneficial to factor the numbers so that you can identify and extract perfect squares, cubes, etc. Doing this makes it possible to simplify the radicals further. Remember to always look for the greatest perfect square factor when simplifying square roots to make the process efficient.

Perfect Square Example:

The number 50 can be factored into \(2\) and \(5^2\). Since \(5^2\) is a perfect square, it can be taken out of the square root when simplifying. You'll be left with \(5\) outside the square root and \(2\) inside, assuming the radicand has no other factors that are perfect squares.

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors, which are the building blocks of all numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself.
To factor a number into its prime components, you can use a factor tree or systematic division by prime numbers. This process is particularly useful when simplifying radicals because it enables you to identify all the squared factors that can exit the radical as single numbers.

Application in Radicals:

In the solution to the provided exercise, prime factorization allows to simplify \(\sqrt{72}\) by expressing it as \(\sqrt{2^3 \times 3^2}\). The squared factor \(3^2\) is then taken out of the radical as \(3\), and what remains is the expression \(6\sqrt{2}\). Prime factorization is a crucial step for simplifying radicals and can also aid in understanding the structure of numbers in various areas of mathematics.