Problem 192 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{2}{3} \sqrt{54}-\frac{3}{4} \sqrt{96} $$

Step-by-Step Solution

Answer

-\sqrt{6}

Step 1 - Simplify the Square Roots

First, simplify each square root individually. For \(\sqrt{54}\), we have \(\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}\). For \(\sqrt{96}\), we have \(\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}\).

Step 2 - Substitute Simplified Square Roots

Substitute the simplified square roots back into the original expression:\(\frac{2}{3} \sqrt{54} - \frac{3}{4} \sqrt{96} = \frac{2}{3}(3\sqrt{6}) - \frac{3}{4}(4\sqrt{6})\).

Step 3 - Simplify the Coefficients

Multiply the coefficients with the square roots:\(\frac{2}{3} \times 3\sqrt{6} = 2\sqrt{6}\), and \(\frac{3}{4} \times 4\sqrt{6} = 3\sqrt{6}\).

Step 4 - Combine Like Terms

Subtract the two terms:\(2\sqrt{6} - 3\sqrt{6} = -\sqrt{6}\).

Key Concepts

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations. In this exercise, we deal with expressions that include square roots. Working with algebraic expressions requires some basic understanding of arithmetic and algebra.

Examples of algebraic expressions include:

• Polynomials, like x² + 2x + 3
• Rational expressions, like (3x + 2)/(x - 1)
• Radical expressions, like \(\sqrt{x} + \sqrt{3x + 5}\)

In this case, we have to simplify expressions that involve radicals. Knowing the properties of square roots and how to manipulate them helps simplify these expressions efficiently. Our goal is to combine like terms, but first, we must simplify the radicals and work through the algebraic steps gradually.

simplifying radicals

Simplifying radicals involves expressing the square root in its simplest form. This often requires the factorization of the number inside the square root.

For example:

• To simplify \(\sqrt{54}\), we recognize that 54 can be factored into 9 and 6. Since 9 is a perfect square (\(9 = 3^2\)), we can write \(\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}\).
• Similarly, with \(\sqrt{96}\), we factor it into 16 and 6. Since 16 is also a perfect square (\(16 = 4^2\)), we get \(96 = 16 \times 6\), and then \(\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}\).

This step allows us to simplify our expression by reducing the radicals to their simplest form, making it easier to perform any algebraic operations required.

combining like terms

Combining like terms is the process of adding or subtracting terms that have the same radical part. In our exercise, we had to deal with terms involving \(\sqrt{6}\). After simplifying the radicals, we substitute them back and then combine the like terms.

• Our original expression is \(\frac{2}{3}\sqrt{54} - \frac{3}{4}\sqrt{96}\).
• After simplification, we substitute back into the expression: \(\frac{2}{3}(3\sqrt{6}) - \frac{3}{4}(4\sqrt{6})\).

Now, multiply the coefficients with the radicals. This gives us \(\frac{2}{3} \times 3\sqrt{6} = 2\sqrt{6}\) and \(\frac{3}{4} \times 4\sqrt{6} = 3\sqrt{6}\).

Finally, we combine the like terms: \(2\sqrt{6} - 3\sqrt{6} = -\sqrt{6}\).

Combining like terms reduces the expression to a simpler form, making it easier to understand and work with. This particular exercise shows the elegance of algebra and the importance of simplifying each part systematically.