Problem 223 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{5}{6} \sqrt{27}+\frac{5}{8} \sqrt{48} $$

Step-by-Step Solution

Answer

The simplified form is \(5 \sqrt{3}\).

Step 1 - Simplify the Radicals

First, simplify \(\frac{5}{6} \sqrt{27}\) and \(\frac{5}{8} \sqrt{48}\). Start by simplifying the square roots. Notice that \(\sqrt{27}\ = \sqrt{9 \cdot 3} = 3 \sqrt{3}\) and \(\sqrt{48}\ = \sqrt{16 \cdot 3} = 4 \sqrt{3}\).

Step 2 - Substitute Simplified Radicals

Next, substitute the simplified radicals back into the expressions: \(\frac{5}{6} \sqrt{27}\) becomes \(\frac{5}{6} \cdot 3 \sqrt{3} = \frac{15}{6} \sqrt{3} = \frac{5}{2} \sqrt{3}\) and \(\frac{5}{8} \sqrt{48}\) becomes \(\frac{5}{8} \cdot 4 \sqrt{3} = \frac{20}{8} \sqrt{3} = 2.5 \sqrt{3}\).

Step 3 - Add the Two Terms

Combine the two terms: \(\frac{5}{2} \sqrt{3} + \frac{20}{8} \sqrt{3}\). Observe that \(\frac{20}{8} = 2.5 = \frac{5}{2}\), so the expression becomes \(\frac{5}{2} \sqrt{3} + \frac{5}{2} \sqrt{3} = 2 \cdot \frac{5}{2} \sqrt{3} = 5 \sqrt{3}\).

Key Concepts

Elementary Algebra

Elementary Algebra involves basic algebraic operations which include addition, subtraction, multiplication, and division of numbers and variables. In this exercise, you are dealing with expressions that need simplification.
Here, you see how algebra applies to handling expressions that involve square roots (radicals).
Remember, practicing these fundamental skills is crucial as they form the basis for more advanced mathematics concepts.

Radical Simplification

Simplifying radicals means rewriting them in their simplest form.
This is usually done by breaking them down into their prime factors. For example, \(\root{27} = \root{9 \cdot 3} = 3 \root{3}\) and \(\root{48} = \root{16 \cdot 3} = 4 \root{3}\). Once simplified, calculations involving these radicals become much easier.
Simplified radicals reveal patterns and relationships that might not be obvious at first glance.

Combining Like Terms

Combining like terms is a fundamental operation in algebra. It involves adding or subtracting terms that have the same variables and exponents.
For this exercise, we combine terms \(\frac{5}{2} \sqrt{3}\) and \(\frac{5}{2} \sqrt{3}\).
Notice both terms have the same radical part (\( \root{3} \)) and coefficients (5/2), making them 'like terms'.
Simplifying becomes straightforward: we just add the coefficients.

Square Roots

Square roots are a type of radical, specifically the \(\root{2}\)-th root of a number.
When you square a number, you get a perfect square, which simplifies neatly back to the original number when square rooted again.
This exercise shows how square roots can be involved in more complex expressions.
Recognizing and simplifying square roots within larger algebraic expressions is a valuable skill for solving problems efficiently.