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Chapter 0: Problem 118

Solve each equation, rounding your answer to four significant digits where necessary. $$ \frac{2}{x^{3}}-\frac{6}{x^{4}}=0 $$

Short Answer

Expert verified
The short answer is: The solution for the given equation \(\frac{2}{x^{3}}-\frac{6}{x^{4}}=0\) is \(x=3\).

Step by step solution

01

Combine the fractions

Since the given equation is: \[ \frac{2}{x^{3}}-\frac{6}{x^{4}}=0, \] our first step is to bring the two fractions on the left side into one fraction. To do this, we need a common denominator. In this case, the least common multiple of the denominators is \(x^{4}\). Thus, we multiply the first fraction by \(\frac{x}{x}\) to match the denominator. Now we have: \[ \frac{2x}{x^{4}}-\frac{6}{x^{4}}=0 \]
02

Combine terms with the same denominator

We can combine the fractions into one since they have the same denominator, \(x^{4}\). This results in: \[ \frac{2x-6}{x^{4}}=0 \]
03

Solve for x by setting the numerator equal to zero

The equation \(\frac{2x-6}{x^4}=0\) holds if and only if the numerator is equal to zero. Therefore, we set the expression \(2x-6\) equal to zero: \[ 2x-6=0 \] Now, we solve for x: \[ 2x=6 \Rightarrow x=3 \] The solution for this equation is \(x=3\). Since there are no decimals in the answer, there is no need to round to four significant digits.

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