Question

the following expression occurs frequently in calculus. simplify the expression.
\frac{sqrt{x}-\frac{1}{10sqrt{x}}}{sqrt{x}}
\frac{sqrt{x}-\frac{1}{10sqrt{x}}}{sqrt{x}}=square\timessquare

Explanation:

Step1: Combine terms in numerator

First, find a common - denominator for the terms in the numerator $\sqrt{x}-\frac{1}{10\sqrt{x}}$. The common denominator is $10\sqrt{x}$, so $\sqrt{x}=\frac{10x}{10\sqrt{x}}$, and $\sqrt{x}-\frac{1}{10\sqrt{x}}=\frac{10x - 1}{10\sqrt{x}}$.

Step2: Divide by $\sqrt{x}$

Now, we have $\frac{\frac{10x - 1}{10\sqrt{x}}}{\sqrt{x}}$. When dividing by a number is the same as multiplying by its reciprocal, so $\frac{\frac{10x - 1}{10\sqrt{x}}}{\sqrt{x}}=\frac{10x - 1}{10\sqrt{x}}\times\frac{1}{\sqrt{x}}=\frac{10x - 1}{10x}$.
We can rewrite $\frac{10x - 1}{10x}$ as $\frac{10x}{10x}-\frac{1}{10x}=1-\frac{1}{10x}= \frac{1}{1}\times x^{0}-\frac{1}{10}\times x^{- 1}$.

Answer:

$1-\frac{1}{10x}$