Simplify (9x^3y^4)÷(x^2)y^2
The problem is to perform algebraic simplification on a given rational expression. Specifically, the question is asking to simplify the expression (9x^3y^4) divided by (x^2)y^2 by properly applying the laws of exponents and performing any necessary division of coefficients and variables. The main task is to reduce the expression to its simplest form by canceling out any common factors in the numerator and the denominator, and by correctly subtracting the exponents of like bases according to the properties of exponents.
$9 x^{3} y^{4} \div x^{2} y^{2}$
Answer
Solution:
Step 1:
Express the division as a fraction: $\frac{9x^3y^4}{x^2y^2}$.
Step 2:
Identify and eliminate common factors between the numerator and the denominator.
Step 2.1:
Extract $x^2$ from the numerator: $\frac{x^2(9xy^4)}{x^2y^2}$.
Step 2.2:
Proceed to simplify by removing common factors.
Step 2.2.1:
Extract $x^2$ from the denominator: $\frac{x^2(9xy^4)}{x^2(y^2)}$.
Step 2.2.2:
Eliminate the $x^2$ term present in both numerator and denominator: $\frac{\cancel{x^2}(9xy^4)}{\cancel{x^2}y^2}$.
Step 2.2.3:
Simplify the fraction: $\frac{9xy^4}{y^2}$.
Step 3:
Further reduce the fraction by canceling out additional common factors.
Step 3.1:
Factor out $y^2$ from the numerator: $\frac{y^2(9xy^2)}{y^2}$.
Step 3.2:
Continue simplification by canceling out common factors.
Step 3.2.1:
Represent the denominator as a product with 1: $\frac{y^2(9xy^2)}{y^2 \cdot 1}$.
Step 3.2.2:
Remove the common $y^2$ factor: $\frac{\cancel{y^2}(9xy^2)}{\cancel{y^2} \cdot 1}$.
Step 3.2.3:
Express the simplified fraction: $\frac{9xy^2}{1}$.
Step 3.2.4:
Divide the numerator by 1 to get the final result: $9xy^2$.
Knowledge Notes:
To simplify a fraction involving algebraic expressions, follow these steps:
Fraction Notation: Begin by writing the division as a fraction, which is a more convenient form for identifying common factors and simplifying.
Factorization: Break down both the numerator and the denominator into their prime factors or common algebraic factors.
Cancellation: If the same factor appears in both the numerator and the denominator, you can cancel it out because dividing a number by itself yields 1.
Exponent Rules: When dealing with variables raised to a power, remember the law of exponents which states that $x^a / x^b = x^{a-b}$ when $a > b$. This is used to simplify expressions with the same base.
Simplify: After canceling common factors, rewrite the fraction to reflect the simplified form. If the denominator is 1, you can omit it, as any number divided by 1 is the number itself.
Final Expression: The result after all simplification steps should be an algebraic expression with no common factors between the numerator and the denominator, and it should be in its simplest form.
