Problem 7 - Algebra for College Students

Question

Multiply and simplify where possible. \((-3 \sqrt{3})(-4 \sqrt{8})\)

Step-by-Step Solution

Answer

The simplified product is \(24\sqrt{6}\).

Step 1: Multiply the coefficients

First, let's multiply the coefficients of the radicals together. The coefficients in the expression \((-3 \sqrt{3})(-4 \sqrt{8})\) are -3 and -4. Multiply these two coefficients: \(-3 \times -4 = 12\).

Step 2: Multiply the radicands

Next, we multiply the radicands themselves, which are under the square root signs. In this case, we multiply \(\sqrt{3}\) by \(\sqrt{8}\):\(\sqrt{3} \times \sqrt{8} = \sqrt{24}\).

Step 3: Simplify the square root

The next step is to simplify \(\sqrt{24}\). Recognize that \(24\) can be factored into \(4 \times 6\), and the square root of \(4\) is \(2\). This gives us:\(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\).

Step 4: Combine all the results

Now combine the results from Step 1 and Step 3 to get the final answer. Multiply \(12\) (from Step 1) by \(2\sqrt{6}\) (from Step 3):\(12 \times 2\sqrt{6} = 24\sqrt{6}\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form. A radical is any mathematical expression that includes a root symbol, often a square root. The process can be boiled down into breaking the number under the radical into factors, searching for perfect squares. For instance, when simplifying \( \sqrt{24} \), you need to express 24 in terms of factors that include a perfect square. In this case, 24 can be broken down into \(4 \times 6\).

• The perfect square \(4\) can come out of the radical as \(2\), because \(\sqrt{4} = 2\).
• The expression \(\sqrt{24}\) thus becomes \(2\sqrt{6}\).

This transformation allows the radical to be more easily combined with other terms or processed in further mathematical operations.

Coefficients and Radicands

In the context of algebraic multiplication involving radicals, it's important to differentiate between coefficients and radicands. The coefficient is the number outside the radical sign, while the radicand is the number inside the radical sign.

• The expression \(-3 \sqrt{3}\) has a coefficient of \(-3\) and a radicand of \(3\).
• Similarly, \(-4 \sqrt{8}\) has a coefficient of \(-4\) and a radicand of \(8\).

To multiply such expressions, we handle the coefficients and the radicands separately:

• First, multiply the coefficients: \(-3\) and \(-4\) give \(12\).
• Then, multiply the radicands: \(\sqrt{3}\) and \(\sqrt{8}\) produce \(\sqrt{24}\).

Understanding the distinction and separate operations on coefficients and radicands is crucial for simplifying these expressions correctly.

Square Roots in Algebra

Square roots frequently appear in algebraic expressions, requiring a good understanding of how to work with them. A square root essentially "undoes" squaring, leading to the value which, when multiplied by itself, gives the original number under the radical. In algebra, squares and square roots help solve various equations, simplify expressions, and express values in a more tractable form.

• For example, given \( \sqrt{9} = 3\), this tells us that 3 squared returns 9.
• This concept extends to simplifying expressions such as radicals, as seen in \( \sqrt{24} = 2 \sqrt{6}\).

When dealing with complex algebraic expressions, managing square roots effectively allows for step-by-step simplification, as illustrated in the original exercise. Once the radicals are simplified and coefficients multiplied, the expression becomes easier to interpret and physically compute.