Problem 24 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ 3 \sqrt{8 x}(2 \sqrt{2 x^{3} y}) $$

Step-by-Step Solution

Answer

The expression simplifies to \( 12x^2 \sqrt{y} \).

Step 1: Understanding the Problem

We need to multiply two expressions involving square roots: \( 3 \sqrt{8x} \) and \( 2 \sqrt{2x^3y} \). We will simplify the product after multiplying.

Step 2: Simplify Each Square Root

Break down each square root into its simplest form. For \( \sqrt{8x} \), this becomes \( \sqrt{4 \cdot 2x} = 2\sqrt{2x} \). For \( \sqrt{2x^3y} \), this can be simplified by factoring \( x^3 \) as \( x^2 \cdot x \), resulting in \( x\sqrt{2xy} \). Thus, \( 2 \sqrt{2x^3y} = 2x\sqrt{2xy} \).

Step 3: Multiply Numerical Coefficients

Multiply the coefficients outside the square roots: \( 3 \times 2 = 6 \) and also account for \( 6 \times x = 6x \). This becomes the coefficient multiplying the simplified square roots.

Step 4: Multiply the Simplified Square Roots

Multiply the square root parts: \( \sqrt{2x} \times \sqrt{2xy} = \sqrt{4x^2y} \). Since \( \sqrt{4x^2y} = 2x\sqrt{y} \) (because \( \sqrt{4} = 2 \) and \( \sqrt{x^2} = x \)), the result simplifies to \( 2x\sqrt{y} \).

Step 5: Combine All Parts

Combining the coefficient from Step 3 with the simplified part from Step 4, we get: \( 6x \times 2x \sqrt{y} = 12x^2 \sqrt{y} \).

Step 6: Final Simplified Result

The full expression \( 3 \sqrt{8x}(2 \sqrt{2x^3y}) \) simplifies to \( 12x^2 \sqrt{y} \).

Key Concepts

Square Roots

The concept of square roots is a fundamental element when simplifying expressions involving radicals. A square root is a value that, when multiplied by itself, yields the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. In radical expressions, square roots are often represented by the symbol \( \sqrt{} \), and they play a vital role in simplifying these expressions.

• The process of simplifying a square root involves breaking down the number inside the radical to its prime factors.
• When a perfect square factor is identified, it can be taken out of the square root.
• For instance, \( \sqrt{8} \) can be rewritten as \( \sqrt{4 \cdot 2} = 2\sqrt{2} \), because \( \sqrt{4} = 2 \).

The ability to simplify square roots is crucial for solving algebraic expressions that contain radical terms.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In the given exercise, the expression involves both numerical coefficients and variables. Variables represent unknown values and are often denoted using letters such as \( x \) and \( y \). Understanding algebraic expressions is a key skill in simplifying terms and solving equations.

• An algebraic expression can include terms that are combined using addition, subtraction, multiplication, and division.
• Within an expression, like \( 3 \sqrt{8x} \), the 3 is a coefficient, while \( 8x \) is a product of a number and a variable.
• Simplifying involves ensuring that each term is as simple as possible and combining like terms when applicable.

Algebraic manipulation, including factoring and simplifying, is essential for simplifying expressions and solving problems effectively.

Multiplication of Radicals

Multiplying radicals is a crucial skill in dealing with expressions that contain square roots and other radicals. When multiplying radicals, we use the property that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b} \). This property allows us to combine and simplify radical expressions effectively.

• When multiplying two radical expressions, first multiply their coefficients (numbers outside the radicals).
• Next, multiply the numbers or variables inside the radicals together.
• For instance, \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \).

In our original exercise, multiplying radicals of \( \sqrt{2x} \) and \( \sqrt{2xy} \) results in \( \sqrt{4x^2y} \), which can be further simplified. Understanding this process allows you to handle more complex expressions and achieve the simplest form efficiently.