Divide (18r)÷3r^4
The question asks to perform a division operation on a monomial expression. You are provided with the expression (18r) and instructed to divide it by another monomial, 3r^4. The task involves simplifying the expression by dividing the coefficients (numerical parts) and subtracting the exponents of the like bases, according to the rules of exponents for division.
$18 r \div 3 r^{4}$
Answer
Solution:
Step 1:
Convert the division operation into a fraction notation: $\frac{18r}{3r^4}$.
Step 2:
Identify and simplify the common factors between the numerator and denominator.
Step 2.1:
Extract the factor of $3$ from $18r$: $\frac{3(6r)}{3r^4}$.
Step 2.2:
Proceed to eliminate the common factors.
Step 2.2.1:
Extract the factor of $3$ from $3r^4$: $\frac{3(6r)}{3(r^4)}$.
Step 2.2.2:
Eliminate the common factor of $3$: $\frac{\cancel{3}(6r)}{\cancel{3}r^4}$.
Step 2.2.3:
Simplify the expression: $\frac{6r}{r^4}$.
Step 3:
Further reduce the fraction by canceling out the common variable factors.
Step 3.1:
Separate the factor of $r$ from $6r$: $\frac{r \cdot 6}{r^4}$.
Step 3.2:
Continue with the cancellation of common variable factors.
Step 3.2.1:
Separate the factor of $r$ from $r^4$: $\frac{r \cdot 6}{r \cdot r^3}$.
Step 3.2.2:
Eliminate the common factor of $r$: $\frac{\cancel{r} \cdot 6}{\cancel{r} \cdot r^3}$.
Step 3.2.3:
Finalize the simplification: $\frac{6}{r^3}$.
Knowledge Notes:
The problem involves simplifying a rational expression, which is a fraction where both the numerator and the denominator are polynomials. The steps taken to simplify the expression are based on the following knowledge points:
Division as Fraction: Division between two algebraic expressions can be represented as a fraction, where the dividend (the number being divided) is the numerator, and the divisor (the number by which the dividend is divided) is the denominator.
Common Factors: When simplifying fractions, any common factors in the numerator and denominator can be canceled out. This is based on the property that $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$, provided $c \neq 0$.
Factoring: Factoring involves rewriting an expression as a product of its factors. This can make it easier to identify and cancel common factors.
Variable Cancellation: When variables are raised to a power, they can be canceled in a similar manner to numerical factors. For example, $r$ in the numerator can cancel with one $r$ in $r^4$ in the denominator, leaving $r^3$.
Simplification: The goal is to rewrite the expression in its simplest form, where no further common factors can be canceled between the numerator and the denominator.
By applying these principles, the given algebraic expression is simplified step by step, ensuring that each action adheres to the rules of algebraic operations.
