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 6x4 as 2(3x4) to reveal the common factor: frac2(3x4)2x6. 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 2x6: frac2(3x4)2(x6). - Step 1.2.2: Cancel out the 2: fraccancel2(3x4)cancel2x6. - Step 1.2.3: Present the simplified expression: frac3x4x6. 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 3x4 as x4 cdot 3: fracx4 cdot 3x6. Step 2.2: Eliminate the Common Variable Factors - Remove the common variable factor from the fraction. - Step 2.2.1: Factor out x4 from x6: fracx4 cdot 3x4 cdot x2. - Step 2.2.2: Cancel out x4: fraccancelx4 cdot 3cancelx4 cdot x2. - Step 2.2.3: Present the final simplified expression: frac3x2. Knowledge Notes: To simplify a fraction involving algebraic expressions, follow these steps: 1. Factorization: Break down both the numerator and the denominator into their prime factors, including numerical coefficients and variables. 2. 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 fracaa = 1 for any non-zero a. 3. 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: xm / xn = xm-n). 4. Rewrite the Expression: After canceling, rewrite the expression to reflect the simplified form. 5. Check for Further Simplification: Ensure that no further common factors can be canceled. …

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

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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 $6 x^{4}$ as $2 (3 x^{4})$ to reveal the common factor: $\frac{2 (3 x^{4})}{2 x^{6}}$ .

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 $2 x^{6}$ : $\frac{2 (3 x^{4})}{2 (x^{6})}$ .
- Step 1.2.2: Cancel out the $2$ : $\frac{2 (3 x^{4})}{2 x^{6}}$ .
- Step 1.2.3: Present the simplified expression: $\frac{3 x^{4}}{x^{6}}$ .

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 $3 x^{4}$ 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 $x^{4}$ : $\frac{x^{4} \cdot 3}{x^{4} \cdot x^{2}}$ .
- 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.