Simplify the Radical Expression x^2y^2 square root of 63x^2y^6
In this problem, you are asked to simplify a mathematical expression that includes a radical, specifically a square root. The expression to be simplified is the product of a variable term, x^2y^2, and the square root of another term, 63x^2y^6. Simplifying a radical expression typically involves breaking down the term inside the radical into its prime factors, if possible, and then simplifying by bringing any factors that are perfect squares out of the radical. This will often also include simplifying any coefficients and like terms both inside and outside of the radical.
$x^{2} y^{2} \sqrt{63 x^{2} y^{6}}$
Answer
Solution:
Step:1
Express $63x^{2}y^{6}$ as $(3xy^{3})^{2} \cdot 7$.
Step:1.1
Extract $9$ from $63$. $x^{2}y^{2}\sqrt{9(7)x^{2}y^{6}}$
Step:1.2
Represent $9$ as $3^{2}$. $x^{2}y^{2}\sqrt{3^{2} \cdot 7x^{2}y^{6}}$
Step:1.3
Express $y^{6}$ as $(y^{3})^{2}$. $x^{2}y^{2}\sqrt{3^{2} \cdot 7x^{2}(y^{3})^{2}}$
Step:1.4
Reposition $7$. $x^{2}y^{2}\sqrt{3^{2}x^{2}(y^{3})^{2} \cdot 7}$
Step:1.5
Represent $3^{2}x^{2}(y^{3})^{2}$ as $(3xy^{3})^{2}$. $x^{2}y^{2}\sqrt{(3xy^{3})^{2} \cdot 7}$
Step:2
Extract terms from under the radical. $x^{2}y^{2}(3xy^{3}\sqrt{7})$
Step:3
Apply the commutative property of multiplication. $\sqrt{7}x^{2}y^{2}(3xy^{3})$
Step:4
Eliminate non-negative terms from the absolute value. $\sqrt{7}x^{2}y^{2} \cdot 3xy^{3}$
Knowledge Notes:
Radical Expressions: A radical expression is an expression that includes a square root, cube root, or any other root. Simplifying a radical expression involves rewriting it in a simpler or more explicit form without changing its value.
Factoring: Factoring is the process of breaking down a number or expression into its constituent factors. In this case, $63$ is factored into $9 \cdot 7$, where $9$ can be further expressed as $3^{2}$.
Square Roots and Exponents: The square root of a number is a value that, when multiplied by itself, gives the original number. For exponents, $(a^{m})^{n} = a^{mn}$. In this problem, $y^{6}$ is rewritten as $(y^{3})^{2}$, which is a property of exponents.
Simplifying Under the Radical: When simplifying expressions under a radical, any perfect square factors can be taken out of the radical, which is what happens when $(3xy^{3})^{2}$ is moved outside of the square root as $3xy^{3}$.
Absolute Value: The absolute value of a number is its non-negative value. Since $x$ and $y$ are not specified as positive or negative, we use absolute value bars to indicate that the expression outside the radical should be non-negative. However, if it is known that the variables represent non-negative quantities, the absolute value bars can be omitted.
Commutative Property of Multiplication: This property states that the order of multiplication does not affect the product. In other words, $ab = ba$. This property is used to rearrange the terms for the final simplified form.
By understanding these concepts, one can simplify radical expressions and manipulate algebraic expressions effectively.
