Problem 60 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{18 x^{2} y} \sqrt{2 x^{2} y} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $$6x^2y$$.

Step 1: Find the square root of the coefficients

: In the given expression, the coefficients are 18 and 2. We need to find the square root of their product. So, we have $$ \sqrt{18 \times 2} = \sqrt{36} = 6. $$

Step 2: Find the square root of the variables

: Now, let's find the square root of the variables, which are \(x^2\) and \(y\). The square root of \(x^2\) is \(x\), and the square root of \(y\) is \(\sqrt{y}\).

Step 3: Multiply the simplified square roots

: We'll now multiply the square roots we found in Steps 1 and 2. So, $$ \sqrt{18x^2y}\hspace{1mm}\sqrt{2x^2y} = 6x\sqrt{y} \cdot x\sqrt{y}. $$

Step 4: Simplify the expression

: Finally, we'll simplify the expression obtained in Step 3 by multiplying the terms: $$ 6x\sqrt{y} \cdot x\sqrt{y} = 6x^2y. $$ Therefore, the expression in its simplified form is $$ \sqrt{18x^2y}\hspace{1mm}\sqrt{2x^2y} = 6x^2y. $$

Key Concepts

Square Roots

Square roots are a fundamental aspect of algebra that help us find a number which, when multiplied by itself, results in the given number. For example, the square root of 36 is 6, because 6 times 6 is 36. Mathematically, this is shown as \( \sqrt{36} = 6 \). When dealing with variables, the square root works similarly. If you have a term like \( x^2 \), you can find its square root by dividing the power by 2, resulting in \( x \). This is why finding the square roots of algebraic expressions involves simplifying both numbers and variables separately. By understanding how square roots function, you can simplify complex mathematical problems more effectively.

Multiplication of Radicals

Multiplying radicals involves a few straightforward rules. Radicals are numbers or expressions under a square root sign. When you're multiplying them, you should do so in steps.

• First, multiply the numbers inside the square root signs together.
• Then, simplify the resulting square root, if possible.

For instance, considering the expression \( \sqrt{18} \times \sqrt{2} \), the multiplication inside the radical gives us \( \sqrt{18 \times 2} = \sqrt{36} \).This equals 6 because 6 multiplied by itself yields 36. This approach allows for more orderly and understandable calculations, whether it involves integers or algebraic expressions.

Simplification Steps

Simplification is the process of reducing an expression to its simplest form. This involves breaking down each part of the expression and combining them as efficiently as possible. In the given problem:1. We first computed the square root of the coefficients, which were 18 and 2. The product is 36, and its square root is 6. 2. Next, we calculated the square root of the variables. Since \( x^2 \) becomes \( x \), and \( y \) remains under the square root as \( \sqrt{y} \), it was simplified to \( 6x\sqrt{y} \). 3. The final step was to multiply these simplified parts: \( 6x\sqrt{y} \times x\sqrt{y} = 6x^2y \).By following the step-by-step approach to simplify each component, you reach an expression that's much easier to work with or interpret.