Problem 210 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{2}{3} \sqrt{27}+\frac{3}{4} \sqrt{48} $$

Step-by-Step Solution

Answer

5\sqrt{3}

Step 1 - Simplify \(\frac{2}{3} \sqrt{27}\)

First, simplify \(\frac{2}{3} \sqrt{27}\) by breaking down the square root. \(\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}\). Therefore, \(\frac{2}{3} \sqrt{27}\) becomes \(\frac{2}{3} \cdot 3\sqrt{3} = 2\sqrt{3}\).

Step 2 - Simplify \(\frac{3}{4} \sqrt{48}\)

Next, simplify \(\frac{3}{4} \sqrt{48}\). Break down the square root: \(\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\). Thus, \(\frac{3}{4} \sqrt{48}\) becomes \(\frac{3}{4} \cdot 4\sqrt{3} = 3\sqrt{3}\).

Step 3 - Combine Like Terms

Combine the simplified terms from Step 1 and Step 2. We have \(\frac{2}{3} \sqrt{27} + \frac{3}{4} \sqrt{48} = 2\sqrt{3} + 3\sqrt{3}\). Adding these together gives \((2 + 3)\sqrt{3} = 5\sqrt{3}\).

Key Concepts

square roots

Square roots are a way of finding a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9.
If you see the symbol \( \sqrt{} \), it indicates the square root. Breaking down numbers into their square roots can simplify expressions.
For instance, \( \sqrt{27} \) can be written as \( \sqrt{9 \times 3} \), which then becomes \( 3\sqrt{3} \), making it easier to work with in algebraic expressions.

radical expressions

Radical expressions involve roots, such as square roots, and are used often in algebra. They follow rules similar to numbers but often require simplification. For example, in the exercise above, we had to simplify \( \sqrt{27} \) and \( \sqrt{48} \).
Here's how:

• First, break down the numbers under the root into their prime factors or recognizable squares.
• For example, \( \sqrt{27} = \sqrt{9 \times 3} \), and because \( \sqrt{9} = 3 \), this simplifies to \( 3\sqrt{3} \).

Simplification makes it easier to combine and work with these expressions.

algebraic simplification

Algebraic simplification involves reducing complex expressions into simpler forms. This makes equations easier to solve or understand.
In the exercise provided, we simplified \( \frac{2}{3}\sqrt{27} \) and \( \frac{3}{4} \sqrt{48} \).
We turned \( \sqrt{27} \) into \( 3\sqrt{3} \). Multiplying this by \( \frac{2}{3} \), we get \( 2\sqrt{3} \).
Similarly, \( \sqrt{48} = 4\sqrt{3} \), and multiplying this by \( \frac{3}{4} \), we get \( 3\sqrt{3} \). Now, these are easier to work with.

combining like terms

Combining like terms is an essential part of algebra. Like terms share the same variables or radicals, making them easy to add or subtract.
In the example above, we had \( 2\sqrt{3} \) and \( 3\sqrt{3} \). Both terms have \( \sqrt{3} \), so we can combine them.
Adding \( 2\sqrt{3} \) and \( 3\sqrt{3} \) gives us \( 5\sqrt{3} \).
This simplification process reduces complexity and makes solving equations straightforward. Remember, only like terms can be combined directly.