Simplify (-10d^4e^3f^5)/(15d^2e^2f)
The given problem is asking to simplify a mathematical expression that involves the division of two algebraic terms, each consisting of a numerical coefficient and variables raised to certain powers. The operation to be performed is to simplify the expression by dividing the numerical coefficients and subtract the exponents of the corresponding variables with like bases in the numerator and the denominator according to the laws of exponents. This involves reducing the fraction to its simplest form by canceling out common factors and applying the exponent rules for division.
$\frac{- 10 d^{4} e^{3} f^{5}}{15 d^{2} e^{2} f}$
Answer
Solution:
Step 1:
Identify and simplify the common numerical factor between $-10$ and $15$.
Step 1.1:
Extract the factor of $5$ from $-10d^4e^3f^5$ to get $\frac{5(-2d^4e^3f^5)}{15d^2e^2f}$.
Step 1.2:
Proceed to reduce the common numerical factors.
Step 1.2.1:
Extract the factor of $5$ from $15d^2e^2f$ to obtain $\frac{5(-2d^4e^3f^5)}{5(3d^2e^2f)}$.
Step 1.2.2:
Eliminate the common factor of $5$ resulting in $\frac{\cancel{5}(-2d^4e^3f^5)}{\cancel{5}(3d^2e^2f)}$.
Step 1.2.3:
Restate the simplified expression as $\frac{-2d^4e^3f^5}{3d^2e^2f}$.
Step 2:
Reduce the common variable factor between $d^4$ and $d^2$.
Step 2.1:
Factor out $d^2$ from $-2d^4e^3f^5$ to get $\frac{d^2(-2d^2e^3f^5)}{3d^2e^2f}$.
Step 2.2:
Proceed to cancel out the common variable factors.
Step 2.2.1:
Factor out $d^2$ from $3d^2e^2f$ to obtain $\frac{d^2(-2d^2e^3f^5)}{d^2(3e^2f)}$.
Step 2.2.2:
Eliminate the common factor of $d^2$ resulting in $\frac{\cancel{d^2}(-2d^2e^3f^5)}{\cancel{d^2}(3e^2f)}$.
Step 2.2.3:
Restate the simplified expression as $\frac{-2d^2e^3f^5}{3e^2f}$.
Step 3:
Reduce the common variable factor between $e^3$ and $e^2$.
Step 3.1:
Factor out $e^2$ from $-2d^2e^3f^5$ to get $\frac{e^2(-2d^2ef^5)}{3e^2f}$.
Step 3.2:
Proceed to cancel out the common variable factors.
Step 3.2.1:
Factor out $e^2$ from $3e^2f$ to obtain $\frac{e^2(-2d^2ef^5)}{e^2(3f)}$.
Step 3.2.2:
Eliminate the common factor of $e^2$ resulting in $\frac{\cancel{e^2}(-2d^2ef^5)}{\cancel{e^2}(3f)}$.
Step 3.2.3:
Restate the simplified expression as $\frac{-2d^2ef^5}{3f}$.
Step 4:
Reduce the common variable factor between $f^5$ and $f$.
Step 4.1:
Factor out $f$ from $-2d^2ef^5$ to get $\frac{f(-2d^2ef^4)}{3f}$.
Step 4.2:
Proceed to cancel out the common variable factors.
Step 4.2.1:
Factor out $f$ from $3f$ to obtain $\frac{f(-2d^2ef^4)}{f \cdot 3}$.
Step 4.2.2:
Eliminate the common factor of $f$ resulting in $\frac{\cancel{f}(-2d^2ef^4)}{\cancel{f} \cdot 3}$.
Step 4.2.3:
Restate the simplified expression as $\frac{-2d^2ef^4}{3}$.
Step 5:
Position the negative sign in front of the fraction to finalize the expression as $-\frac{2d^2ef^4}{3}$.
Knowledge Notes:
The problem involves simplifying a rational expression by canceling out common factors in the numerator and the denominator. The process requires understanding of the following concepts:
Factorization: The process of breaking down numbers or expressions into their constituent factors, which when multiplied together give the original number or expression.
Common Factors: These are factors that are the same in both the numerator and the denominator of a fraction. They can be numerical or variable factors.
Cancellation: When a factor appears in both the numerator and the denominator, it can be 'cancelled out' or reduced to 1, simplifying the expression.
Exponent Rules: When variables have exponents, the rules of exponents apply. For example, $a^m / a^n = a^{m-n}$ when $m > n$.
Negative Signs: A negative sign can be moved in front of a fraction or to the numerator or denominator, but it must be accounted for in the final expression.
Simplifying Rational Expressions: The process of reducing expressions to their simplest form by canceling common factors.
LaTeX Typesetting: A typesetting system that is widely used for mathematical and scientific documents, due to its powerful handling of formulas and bibliographies. In this context, LaTeX is used to format mathematical expressions for clarity and precision.
