Multiply. Simplify if possible.
[tex]\frac{14 x}{2} \cdot \frac{x^7}{7 x^5}[/tex]
Answer (2)
To simplify the expression 2 14 x ⋅ 7 x 5 x 7 , we multiply and simplify to arrive at x 3 . The process involves multiplying the numerators and denominators, reducing coefficients, and simplifying the variable terms. The final simplified result is x 3 .
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Multiply the numerators and denominators: 2 ⋅ 7 x 5 14 x ⋅ x 7 = 14 x 5 14 x 8 .
Simplify the coefficients: 14 14 = 1 .
Simplify the variable terms using the exponent rule: x 5 x 8 = x 8 − 5 = x 3 .
The final simplified expression is: x 3 .
Explanation
Initial Expression First, we need to multiply the given expression and simplify it. The expression is:
Expression in LaTeX 2 14 x ⋅ 7 x 5 x 7
Multiplying Fractions Multiply the numerators and the denominators:
Combined Fraction 2 ⋅ 7 x 5 14 x ⋅ x 7 = 14 x 5 14 x 8
Simplifying Coefficients Now, simplify the fraction by canceling out the common factors. First, we simplify the coefficients:
Coefficient Simplification 14 14 = 1
Simplifying Variables Next, we simplify the variable terms using the exponent rule x b x a = x a − b :
Variable Simplification x 5 x 8 = x 8 − 5 = x 3
Final Result Therefore, the simplified expression is:
Simplified Expression x 3
Examples
Understanding how to simplify expressions like this is useful in many areas, such as physics and engineering, where you often deal with complex formulas. For example, when calculating the trajectory of a projectile, you might need to simplify an expression involving variables like time and distance. Simplifying the expression makes it easier to understand the relationship between these variables and to make accurate predictions. This skill is also crucial in computer science when optimizing algorithms and reducing computational complexity.
