Simplify (6x^4)/(2x^6)
The question is asking you to perform the algebraic operation of simplification on the given rational expression, which is a fraction that consists of two polynomials. Specifically, the question requires you to simplify a fraction where the numerator is '6' multiplied by the variable 'x' raised to the power of '4' (6x^4), and the denominator is '2' multiplied by the variable 'x' raised to the power of '6' (2x^6). Simplifying this expression involves reducing it to its simplest form using algebraic rules concerning exponents and division of like terms.
$\frac{6 x^{4}}{2 x^{6}}$
Answer
Solution:
Step 1: Simplify the Numerical Coefficients
- Identify and divide out the common numerical factor from the numerator and denominator.
Step 1.1: Factor Out the Common Numerical Coefficient
- Express $6x^4$ as $2(3x^4)$ to reveal the common factor: $\frac{2(3x^4)}{2x^6}$.
Step 1.2: Eliminate the Common Numerical Factors
Remove the common numerical factor from the fraction.
Step 1.2.1: Factor out the $2$ from $2x^6$: $\frac{2(3x^4)}{2(x^6)}$.
Step 1.2.2: Cancel out the $2$: $\frac{\cancel{2}(3x^4)}{\cancel{2}x^6}$.
Step 1.2.3: Present the simplified expression: $\frac{3x^4}{x^6}$.
Step 2: Simplify the Variable Factors
- Cancel out the common variable factors from the numerator and denominator.
Step 2.1: Factor Out the Common Variable Factor
- Write $3x^4$ as $x^4 \cdot 3$: $\frac{x^4 \cdot 3}{x^6}$.
Step 2.2: Eliminate the Common Variable Factors
Remove the common variable factor from the fraction.
Step 2.2.1: Factor out $x^4$ from $x^6$: $\frac{x^4 \cdot 3}{x^4 \cdot x^2}$.
Step 2.2.2: Cancel out $x^4$: $\frac{\cancel{x^4} \cdot 3}{\cancel{x^4} \cdot x^2}$.
Step 2.2.3: Present the final simplified expression: $\frac{3}{x^2}$.
Knowledge Notes:
To simplify a fraction involving algebraic expressions, follow these steps:
Factorization: Break down both the numerator and the denominator into their prime factors, including numerical coefficients and variables.
Cancel Common Factors: If the same factor appears in both the numerator and the denominator, you can cancel it out. This is based on the property that $\frac{a}{a} = 1$ for any non-zero $a$.
Apply Exponent Rules: When variables have exponents and you're dividing them, subtract the exponents if the base is the same (according to the quotient rule: $x^m / x^n = x^{m-n}$).
Rewrite the Expression: After canceling, rewrite the expression to reflect the simplified form.
Check for Further Simplification: Ensure that no further common factors can be canceled. If there are, repeat the process until the expression is fully simplified.
In the given problem, the numerical coefficient 6 in the numerator and 2 in the denominator have a common factor of 2, which is canceled out. Similarly, the variable $x$ with exponents is simplified by subtracting the exponents in the numerator and denominator, resulting in the final simplified expression.
