Simplify square root of 252x*(6y^3)*35
The question asks to simplify an algebraic expression that involves the square root of a product of various terms, including constants and variables. The expression to be simplified is the square root of an entire product: 252x multiplied by 6y^3 multiplied by 35. Simplifying this expression involves factoring, finding perfect squares within the factors, and applying the properties of square roots to the variables involved.
$\sqrt{252 x \cdot \left(\right. 6 y^{3} \left.\right) \cdot 35}$
Answer
Solution:
Step 1:
Multiply the numbers outside the square root. $\sqrt{1512x \cdot 35y^3}$
Step 2:
Combine the multiplication of the constants. $\sqrt{52920xy^3}$
Step 3:
Express $52920xy^3$ as a product of squares and other factors.
Step 3.1:
Extract the square factor from $52920$. $\sqrt{1764 \cdot 30xy^3}$
Step 3.2:
Represent $1764$ as a square of an integer. $\sqrt{(42)^2 \cdot 30xy^3}$
Step 3.3:
Separate the square of $y^2$ from $y^3$. $\sqrt{(42)^2 \cdot 30x(y^2)y}$
Step 3.4:
Rearrange the terms under the radical. $\sqrt{(42)^2 \cdot 30y^2xy}$
Step 3.5:
Reposition the constant $30$. $\sqrt{(42)^2y^2 \cdot 30xy}$
Step 3.6:
Rewrite $(42)^2y^2$ as a square of a product. $\sqrt{((42y)^2) \cdot 30xy}$
Step 3.7:
Enclose the terms in parentheses. $\sqrt{((42y)^2) \cdot (30xy)}$
Step 3.8:
Ensure the expression is correctly parenthesized. $\sqrt{((42y)^2) \cdot (30xy)}$
Step 4:
Extract the square terms from under the square root. $42y\sqrt{30xy}$
Knowledge Notes:
The problem involves simplifying a square root expression that contains both numerical and algebraic factors. The process requires knowledge of several mathematical concepts:
Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a perfect square (like $42^2$) can be simplified to the number itself (in this case, $42$).
Multiplication of Radicals: When multiplying radicals, you can multiply the numbers inside the radicals together and then simplify if possible.
Factoring: Factoring involves rewriting a number or expression as a product of its factors. In this case, we factor $52920$ to find a perfect square factor, which can be taken out of the square root.
Algebraic Manipulation: Rearranging terms and separating variables from constants are common algebraic manipulations to simplify expressions.
Simplifying Expressions: The goal is to write the expression in the simplest form possible. This often involves extracting squares from under the square root and simplifying any remaining radical expression.
The process described in the solution involves these concepts and manipulations to simplify the given square root expression.
