Problem 255 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(4 \sqrt{2 k^{5}}\right)\left(-3 \sqrt{32 k^{6}}\right) $$

Step-by-Step Solution

Answer

-96 k^{11/2}

Step 1: Simplify each square root individually

First, simplify each individual square root expression. Remember that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).\[ 4 \sqrt{2 k^{5}} = 4 \sqrt{2} \sqrt{k^{5}} \]\[ -3 \sqrt{32 k^{6}} = -3 \sqrt{32} \sqrt{k^{6}} \]

Step 2: Simplify the numbers under the square roots

Simplify \sqrt{32} and \sqrt{2}.Since \ 32 = 16 \cdot 2,\[ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4 \sqrt{2} \]Therefore, \[ -3 \sqrt{32} = -3 \cdot 4 \sqrt{2} = -12 \sqrt{2} \]So,\[ \left(4 \sqrt{2} \sqrt{k^{5}}\right) \left(- 12 \sqrt{2} \sqrt{k^{6}}\right) \]

Step 3: Simplify the exponents

Simplify \sqrt{k^{5}} and \sqrt{k^{6}}.Recall that \sqrt{k^{n}} = k^{n/2}.Therefore, \[ \sqrt{k^{5}} = k^{5/2} \text{ and } \sqrt{k^{6}} = k^{6/2} = k^3 \]

Step 4: Multiply the simplified parts

Combine all simplified parts. Multiply the coefficients and then the variables under the same base.\[ \left(4 \sqrt{2} \cdot k^{5/2} \right) \left(- 12 \sqrt{2} \cdot k^{3} \right) \]First combine the coefficients:\[ 4 \cdot -12 \sqrt{2} \cdot \sqrt{2} = -48 \cdot 2 = -96 \]Next combine the variables:\[ k^{5/2} \cdot k^{3} = k^{5/2 + 6/2} = k^{11/2} \]So the expression becomes:\[ -96 k^{11/2} \]

Key Concepts

square roots

Square roots are essential in algebra and appear frequently in various problems. The square root of a number, say \( x \), is a value that, when multiplied by itself, yields \( x \). For instance, the square root of 16 is 4, since \( 4 \times 4 = 16 \). When working with variables inside square roots, remember the formula: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
This comes in handy when simplifying expressions that contain both numbers and variables under the square root, as seen in steps like: \sqrt{32} becoming \sqrt{16 \times 2} = 4 \sqrt{2}. By breaking it down, you make the problem more manageable.

exponents

Understanding exponents is fundamental in simplifying algebraic expressions. An exponent indicates how many times a number, referred to as the base, is multiplied by itself. For instance, \( k^3 \) means \( k \times k \times k \).

When dealing with square roots involving exponents, use the property \( \sqrt{a^n} = a^{n/2} \). For example, \sqrt{k^6} simplifies to \( k^{6/2} = k^3 \). This approach systematically reduces complex expressions. In the given problem, \sqrt{k^5} becomes \( k^{5/2} \), and having this knowledge helps break down expressions efficiently.

multiplication of radicals

Multiplying radicals can initially seem challenging but follows straightforward rules. When multiplying two square roots, you multiply the numbers inside them: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). Each expression needs to be simplified before multiplying, as shown in the breakdown:

• \( 4 \sqrt{2} \times -12 \sqrt{2} = -48 \sqrt{4} \). Since \sqrt{4} = 2, the expression further simplifies to \( -96 \).

This method ensures simplification and accuracy, transforming complex problems into more manageable terms.

simplification steps

Simplifying algebraic expressions involves several steps. This method brings clarity to the problem-solving process. Let's break down each step:

• Simplify individual square roots: This reduces the initial complexity of the problem.
• Utilize exponent rules: Applying \( \sqrt{k^n} = k^{n/2} \) helps in managing the variable parts.
• Combine coefficients and like terms: Multiplying constants and combining exponents of the same base streamlines the expression.

Following these steps in a structured way makes the problem more approachable, leading to the solution \( -96 k^{11/2} \). By consistently following a simplification process, you make complex algebra easier to handle.