Simplify square root of 252x^9y^7
The question is asking to perform a mathematical simplification on the expression under the square root. The expression to be simplified is the square root of a product, which contains a numerical factor (252) and variable factors with exponents (x^9 and y^7). The task involves simplifying this radical by factoring out perfect squares from within the radical and simplifying any powers of variables that may be reduced to lower powers outside the square root, according to the properties of exponents and square roots.
$\sqrt{252 x^{9} y^{7}}$
Answer
Solution:
Step 1: Break down the expression under the square root
Step 1.1: Factor 36 from 252 to get $\sqrt{36 \cdot 7 x^{9} y^{7}}$.
Step 1.2: Express 36 as $6^2$ to have $\sqrt{6^2 \cdot 7 x^{9} y^{7}}$.
Step 1.3: Separate $x^8$ from $x^9$ as $\sqrt{6^2 \cdot 7 (x^8 x) y^{7}}$.
Step 1.4: Represent $x^8$ as $(x^4)^2$ yielding $\sqrt{6^2 \cdot 7 ((x^4)^2 x) y^{7}}$.
Step 1.5: Extract $y^6$ from $y^7$ to form $\sqrt{6^2 \cdot 7 ((x^4)^2 x) (y^6 y)}$.
Step 1.6: Write $y^6$ as $(y^3)^2$ to get $\sqrt{6^2 \cdot 7 ((x^4)^2 x) ((y^3)^2 y)}$.
Step 1.7: Rearrange to place $x$ next to $y$ as $\sqrt{6^2 \cdot 7 ((x^4)^2) (y^3)^2 x y}$.
Step 1.8: Move the 7 outside the squared terms to have $\sqrt{(6^2 ((x^4)^2) (y^3)^2) \cdot 7 x y}$.
Step 1.9: Combine the squared terms under a single square to form $\sqrt{((6 x^4 y^3)^2) \cdot 7 x y}$.
Step 1.10: Enclose the product of 7, x, and y in parentheses to get $\sqrt{((6 x^4 y^3)^2) \cdot (7 x y)}$.
Step 1.11: Ensure the expression is properly parenthesized as $\sqrt{((6 x^4 y^3)^2) \cdot (7 x y)}$.
Step 2: Simplify the square root
- Pull out the squared terms to obtain $6 x^4 y^3 \sqrt{7 x y}$.
Knowledge Notes:
To simplify the square root of an algebraic expression, we follow these steps:
Factorization: Break down the number into its prime factors and express variables with exponents in a way that highlights perfect squares.
Perfect Squares: Recognize that the square root of a perfect square is simply the base of the square. For example, $\sqrt{a^2} = a$.
Simplification: Separate the terms under the square root into perfect squares and the remaining factors.
Extraction: Take the square root of the perfect squares, moving them outside the square root symbol.
Rearrangement: Rearrange the terms if necessary to form a simplified expression.
In this specific problem, we used the following mathematical properties:
The square root of a product is the product of the square roots: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$.
The square root of a square is the base number: $\sqrt{a^2} = a$.
The exponent rule $(a^m)^n = a^{m \cdot n}$, which is useful for simplifying expressions with exponents.
