Simplify square root of 112x^2y

The question asks for the simplification of the mathematical expression which is the square root of "112x^2y". This involves breaking down the expression into its simplest form by factoring out perfect squares from the number 112 and using the property of square roots that allows for the square root of a product to be expressed as the product of the square roots of the individual factors. Additionally, it involves simplifying the square root of the variable expression "x^2y" by identifying any variables raised to powers that are perfect squares.

$\sqrt{112 x^{2} y}$

Answer

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Solution:

Step 1: Express $112 x^{2} y$ in a form that separates perfect squares from other factors.

Step 1.1: Identify the perfect square factor in $112$ . Extract $16$ from $112$ to get $\sqrt{16 \cdot 7 x^{2} y}$ .
Step 1.2: Recognize that $16$ is the square of $4$ , thus write it as $4^{2}$ . This gives us $\sqrt{4^{2} \cdot 7 x^{2} y}$ .
Step 1.3: Rearrange the terms to place similar factors together, resulting in $\sqrt{4^{2} x^{2} \cdot 7 y}$ .
Step 1.4: Realize that $4^{2} x^{2}$ is a perfect square and can be written as $(4 x)^{2}$ . Now we have $\sqrt{(4 x)^{2} \cdot 7 y}$ .
Step 1.5: Enclose the entire expression under the square root in parentheses to emphasize the separate components: $\sqrt{((4 x)^{2} \cdot (7 y))}$ .

Step 2: Simplify the square root by extracting the square terms.

Pull out the square terms from under the square root to get $4 x \sqrt{7 y}$ .

Knowledge Notes:

Square Roots: The square root of a number or expression is a value that, when multiplied by itself, gives the original number or expression. For example, $\sqrt{9} = 3$ because $3 \times 3 = 9$ .
Perfect Squares: These are numbers or expressions that are squares of integers or algebraic terms. For instance, $16$ is a perfect square because it is $4^{2}$ .
Simplifying Square Roots: This involves identifying and extracting perfect square factors from under the square root sign. The square root of a perfect square is simply the number that was squared.
Algebraic Manipulation: Rearranging terms and factoring expressions are common techniques in algebra used to simplify expressions and solve equations.
Radicals: Expressions that involve roots, such as square roots, are called radicals. Simplifying radicals typically involves removing any perfect square factors from under the radical sign.