Problem 306 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(4 \sqrt{12 x^{5}}\right)\left(2 \sqrt{6 x^{3}}\right) $$

Step-by-Step Solution

Answer

48 x^{4} \sqrt{2}

Step 1: Simplify Inside the Radicals

First, simplify inside the square roots. For \(\sqrt{12 x^{5}}\), we get: \[\sqrt{12 x^{5}} = \sqrt{4 \cdot 3 \cdot x^{4} \cdot x} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{x^{4}} \cdot \sqrt{x} = 2 \cdot \sqrt{3} \cdot x^{2} \cdot \sqrt{x} = 2 x^{2} \sqrt{3x}\] For \(\sqrt{6 x^{3}}\), we get: \[\sqrt{6 x^{3}} = \sqrt{6} \cdot \sqrt{x^{2}} \cdot \sqrt{x} = \sqrt{6} \cdot x \cdot \sqrt{x} = x \sqrt{6x}\]

Step 2: Substitute Back the Simplified Expressions

Replace the simplified expressions back into the original problem: \[\left(4 \cdot 2 x^{2}\sqrt{3x}\right) \left(2 \cdot x\sqrt{6x}\right) = 4(2 x^{2} \sqrt{3x}) \cdot 2(x \sqrt{6x})\]

Step 3: Multiply the Terms

Multiply the numerical coefficients and the algebraic terms separately: \[(4 \cdot 2) \cdot (2) \cdot x^{2} \cdot x \sqrt{3x} \cdot \sqrt{6x} = 8 \cdot 2 \cdot x^{3} \cdot \sqrt{18 x^{2}} = 16 x^{3} \sqrt{18 x^{2}}\]

Step 4: Simplify the Radical

Simplify \(\sqrt{18 x^{2}}\)\[\sqrt{18 x^{2}} = \sqrt{9 \cdot 2 \cdot x^{2}} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{x^{2}} = 3 \cdot \sqrt{2} \cdot x\]So, the expression becomes: \[16 x^{3} \cdot 3 x \cdot \sqrt{2} = 48 x^{4} \sqrt{2}\]

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or any other root. In math, a radical is represented by the symbol \( \sqrt{} \). Radicals often contain numbers and variables inside them. To simplify, you need to find factors that are perfect squares (for square roots). In the original exercise, we simplified expressions like \( \sqrt{12 x^{5}} \) and \( \sqrt{6 x^{3}} \) into easier forms. This helps in operations like addition, subtraction, and multiplication.

Algebraic Simplification

Algebraic simplification involves breaking down an expression into its simplest form. This can mean getting rid of parentheses, combining like terms, or simplifying radicals. In the given problem, simplification was done step by step:

• First, we simplified inside each square root by factoring out perfect squares.
• Then, we substituted the simplified expressions back into our main equation.
• Finally, we combined and simplified the numerical and algebraic parts

Simplification is crucial as it makes complex problems more manageable!

Multiplication of Radicals

Multiplying radicals follows specific rules. If you have \( \sqrt{a} \) and \( \sqrt{b} \), you can multiply them to get \( \sqrt{a \cdot b} \). For the exercise, after simplifying the radicals, we combined them:

• First, multiply the outside numbers: \( 4 \cdot 2 \cdot 2 = 16 \)
• Then, multiply the simplified radicals \( \sqrt{3x} \cdot \sqrt{6x} = \sqrt{18 x^{2}} \)

Breaking it into manageable steps makes radical multiplication simpler and more understandable!

Square Roots

Square roots are a special type of radical where the root is 2. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3 \cdot 3 = 9 \).
In our exercise, we relied heavily on square roots, like simplifying \( \sqrt{18 x^{2}} \) by finding perfect squares inside the root. We saw:

• \( \sqrt{18 x^{2}} = \sqrt{9 \cdot 2 \cdot x^{2}} = 3 \sqrt{2}x \)
• This is because \( \sqrt{9} = 3 \) and \( \sqrt{x^{2}} = x \)

Understanding how to simplify square roots is key to mastering radicals.