Simplify the Radical Expression x^2y^2 square root of 63x^2y^6

Solution:

Step:1

Express $63x^2{y}^6$ as $(3x{y}^3{)}^2\cdot 7$.

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 $(3x{y}^3{)}^2$. $x^2{y}^2\sqrt{(3x{y}^3{)}^2\cdot 7}$

Step:2

Extract terms from under the radical. $x^2{y}^2(3x{y}^3\sqrt{7})$

Step:3

Apply the commutative property of multiplication. $\sqrt{7}{x}^2{y}^2(3x{y}^3)$

Step:4

Eliminate non-negative terms from the absolute value. $\sqrt{7}{x}^2{y}^2\cdot 3x{y}^3$

Knowledge Notes:

1. 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.

2. 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 $3^2$.

3. 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.

4. 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 $(3x{y}^3{)}^2$ is moved outside of the square root as $3x{y}^3$.

5. 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.

6. 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.