Problem 15 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$2 \sqrt{75}-4 \sqrt{27}$$

Step-by-Step Solution

Answer

The simplified expression is \(-2\sqrt{3}\).

Step 1: Simplify \(\sqrt{75}\)

First, notice that \(75 = 25 \times 3\). Since \(\sqrt{25} = 5\), we can simplify \(\sqrt{75}\) to \(\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}\).

Step 2: Simplify \(\sqrt{27}\)

Next, observe that \(27 = 9 \times 3\). Since \(\sqrt{9} = 3\), we can simplify \(\sqrt{27}\) to \(\sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\).

Step 3: Substitute Simplified Square Roots

Substitute the simplified square roots back into the original expression to get: \(2(5\sqrt{3}) - 4(3\sqrt{3})\).

Step 4: Distribute the Coefficients

Distribute the coefficients in front of the square roots: \(10\sqrt{3} - 12\sqrt{3}\).

Step 5: Combine Like Terms

Since both terms have \(\sqrt{3}\), combine them: \( (10 - 12) \sqrt{3} = -2\sqrt{3}\).

Key Concepts

Square Roots

Understanding square roots is key when working with expressions like \(2\sqrt{75} - 4\sqrt{27}\). A square root determines what number, when multiplied by itself, results in the given number. For instance, \(\sqrt{9} = 3\) because \(3\times3 = 9\). In simplifying square roots, it's essential to find and factor numbers into perfect squares. Let's take \(75\) as an example:

• Break \(75\) down into \(25\times3\).
• Recognize that \(25\) is a perfect square, since \(\sqrt{25} = 5\).
• Thus, \(\sqrt{75}\) simplifies to \(5\sqrt{3}\).

Similarly, \(\sqrt{27}\) simplifies to \(3\sqrt{3}\). Breaking down a number into perfect squares simplifies the computation and assists further algebraic manipulation, such as combining like terms.

Combining Like Terms

Once square roots are simplified, the next step often involves combining like terms. 'Like terms' are those terms in an expression that have the same variable factors. In the expression \(10\sqrt{3} - 12\sqrt{3}\), both terms contain \(\sqrt{3}\), making them like terms.

• Factor out the common square root \(\sqrt{3}\).
• This gives you \((10 - 12)\sqrt{3}\).
• Simplify to get \(-2\sqrt{3}\).

Combining like terms streamlines expressions by reducing them to their simplest form, easing calculations, and improving readability. It's an essential skill to master in order to handle more complex algebraic expressions efficiently.

Distributive Property

The distributive property is a valuable algebraic principle used extensively in simplifying expressions. This property, expressed as \(a(b+c) = ab + ac\), allows us to multiply terms over an addition or subtraction inside parentheses. It's prominently used here with square roots.
In this problem, after simplifying the square roots, we substitute back into the expression, which looks like \(2(5\sqrt{3}) - 4(3\sqrt{3})\). Applying the distributive property:

• Multiply the \(2\) by \(5\sqrt{3}\) to get \(10\sqrt{3}\).
• Similarly, \(-4\) multiplied by \(3\sqrt{3}\) results in \(-12\sqrt{3}\).

This step is crucial as it organizes expressions by influencing how terms are grouped and simplified. Mastering the distributive property not only simplifies arithmetic operations but also enhances problem-solving skills across various mathematical topics.