Problem 250 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(3 \sqrt{8 c^{5}}\right)\left(2 \sqrt{6 c^{3}}\right) $$

Step-by-Step Solution

Answer

24 \(c^{4} \sqrt{3}\)

Step 1 - Simplify the Product of Constants

First, multiply the constants outside the square roots: \(3 \times 2 = 6\).

Step 2 - Multiply the Radicands

Next, multiply the numbers and variables inside the square roots: \(8 c^{5} \times 6 c^{3} = 48 c^{8}\).

Step 3 - Combine the Results

Combine the result from Step 1 with the square root from Step 2: \(6 \sqrt{48 c^{8}}\).

Step 4 - Simplify the Square Root

Simplify \(\sqrt{48 c^{8}}\). First, factorize 48: \(48 = 16 \times 3\). Then, rewrite the expression: \(\sqrt{48 c^{8}} = \sqrt{16 \times 3 \times c^{8}} = \sqrt{16} \times \sqrt{3} \times \sqrt{c^{8}}\).

Step 5 - Evaluate the Simplified Radicals

Evaluate the square roots: \(\sqrt{16} = 4\) and \(\sqrt{c^{8}} = c^{4}\). So, \(\sqrt{16} \times \sqrt{3} \times \sqrt{c^{8}} = 4 \times c^{4} \times \sqrt{3}\).

Step 6 - Combine All Terms

Combine all the simplified terms: \(6 \times 4 \times c^{4} \times \sqrt{3} = 24 c^{4} \sqrt{3}\).

Key Concepts

Square Roots

Square roots help us find a number which, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. Square roots are often written as \(\sqrt{}\). When handling square roots in algebraic expressions, we need to simplify them by looking for perfect square factors. Perfect squares include numbers like 1, 4, 9, 16, and so on. These simplifications make it easier to work with complex problems.

Exponents

Exponents are shorthand for repeated multiplication. For example, \(c^5\) means \(c \times c \times c \times c \times c\). When working with exponents and radicals together, remember certain rules:

• Multiplying like bases: \(c^x \times c^y = c^{x+y}\).
• Dividing like bases: \(c^x / c^y = c^{x-y}\).

Exponents can also be simplified before taking the square root. For instance, \(c^8\) inside a square root becomes \(c^4 \), because \(\sqrt{c^8} = c^4\). Recognizing these patterns helps simplify expressions more easily.

Multiplication of Radicals

Multiplying radicals involves both the constants outside the radical and the numbers inside. Let's break it down:

• Multiply the constants: In the example, 3 and 2 are outside the square roots, giving \(3 \times 2 = 6\).
• Multiply the numbers inside: Here, we have \(\sqrt{8c^5} \times \sqrt{6c^3} = \sqrt{48c^8}\).

When you multiply radicands (the numbers inside the radical), just combine them directly under one spacious square root. After this, simplify inside the radical to find better consolidated or combined forms.

Simplification

Simplifying radical expressions makes them easier to understand and solve. Let's look at the final steps of simplification:

• Factor the numbers under the radical: Factorize 48 as \(16 \times 3\).
• Break down into separate square roots: \(\sqrt{48c^8} = \sqrt{16} \times \sqrt{3} \times \sqrt{c^8} = 4 \times \sqrt{3} \times c^4\).

Combining everything gives \(24c^4\sqrt{3}\), representing the simplified product. Always look for common factors and perfect squares to make the problem easier.