Full solution

Q. Perform the operation and express your answer as a single fraction in simplest form.

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\frac{1}{5 x^{2}}-6 x

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Answer:

Identify Given Expression: Identify the given expression and the operation to be performed. $\newline$ We are given the expression $(1)/(5x^{2}) - 6x$ and we need to simplify it.
Recognize Type of Problem: Recognize that the expression is not a fraction subtraction problem but rather a polynomial with a fraction and a whole term. $\newline$ We need to find a common denominator to combine the terms into a single fraction.
Determine Least Common Denominator: Determine the least common denominator (LCD) for the terms. $\newline$ The LCD for $5x^2$ and $1$ (considering the term $-6x$ as $-6x/1$ ) is $5x^2$ .
Rewrite Terms with LCD: Rewrite each term with the common denominator. $\newline$ The first term is already over the common denominator, so it remains $(\frac{1}{5x^2})$ . The second term, $-6x$ , needs to be written with the common denominator, which gives us $(\frac{-6x \cdot 5x^2}{5x^2})$ .
Perform Multiplication: Perform the multiplication in the second term. $\newline$ Multiplying $-6x$ by $5x^2$ gives us $-30x^3$ . So now we have $(1)/(5x^2) - (-30x^3)/(5x^2)$ .
Combine Terms Over LCD: Combine the terms over the common denominator. $\newline$ Now we can combine the terms to get a single fraction: $(1 - (-30x^3))/(5x^2)$ .
Simplify Numerator: Simplify the numerator by adding the opposite. $\newline$ The numerator simplifies to $1 + 30x^3$ , so the fraction becomes $(1 + 30x^3)/(5x^2)$ .
Check Further Simplification: Check if the fraction can be simplified further. $\newline$ The fraction $(1 + 30x^3)/(5x^2)$ cannot be simplified further because the numerator and denominator do not have common factors other than $1$ .