Simplify 1/(t square root of 1+t)-1/t

$\frac{1}{t \sqrt{1 + t}} - \frac{1}{t}$

Step 1

Simplify each term.

Multiply $\frac{1}{t \sqrt{1 + t}}$ by $\frac{\sqrt{1 + t}}{\sqrt{1 + t}}$ .

$\frac{1}{t \sqrt{1 + t}} \cdot \frac{\sqrt{1 + t}}{\sqrt{1 + t}} - \frac{1}{t}$

Combine and simplify the denominator.

Multiply $\frac{1}{t \sqrt{1 + t}}$ by $\frac{\sqrt{1 + t}}{\sqrt{1 + t}}$ .

$\frac{\sqrt{1 + t}}{t \sqrt{1 + t} \sqrt{1 + t}} - \frac{1}{t}$

Move $\sqrt{1 + t}$ .

$\frac{\sqrt{1 + t}}{t (\sqrt{1 + t} \sqrt{1 + t})} - \frac{1}{t}$

Raise $\sqrt{1 + t}$ to the power of $1$ .

$\frac{\sqrt{1 + t}}{t ({\sqrt{1 + t}}^{1} \sqrt{1 + t})} - \frac{1}{t}$

Raise $\sqrt{1 + t}$ to the power of $1$ .

$\frac{\sqrt{1 + t}}{t ({\sqrt{1 + t}}^{1} {\sqrt{1 + t}}^{1})} - \frac{1}{t}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{1 + t}}{t {\sqrt{1 + t}}^{1 + 1}} - \frac{1}{t}$

Add $1$ and $1$ .

$\frac{\sqrt{1 + t}}{t {\sqrt{1 + t}}^{2}} - \frac{1}{t}$

Rewrite ${\sqrt{1 + t}}^{2}$ as $1 + t$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + t}$ as ${(1 + t)}^{\frac{1}{2}}$ .

$\frac{\sqrt{1 + t}}{t {({(1 + t)}^{\frac{1}{2}})}^{2}} - \frac{1}{t}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{1 + t}}{t {(1 + t)}^{\frac{1}{2} \cdot 2}} - \frac{1}{t}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{1 + t}}{t {(1 + t)}^{\frac{2}{2}}} - \frac{1}{t}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{1 + t}}{t {(1 + t)}^{\frac{2}{2}}} - \frac{1}{t}$

Rewrite the expression.

$\frac{\sqrt{1 + t}}{t {(1 + t)}^{1}} - \frac{1}{t}$

Simplify.

$\frac{\sqrt{1 + t}}{t (1 + t)} - \frac{1}{t}$

Step 2

To write $- \frac{1}{t}$ as a fraction with a common denominator, multiply by $\frac{1 + t}{1 + t}$ .

$\frac{\sqrt{1 + t}}{t (1 + t)} - \frac{1}{t} \cdot \frac{1 + t}{1 + t}$

Step 3

Simplify terms.

Multiply $\frac{1}{t}$ by $\frac{1 + t}{1 + t}$ .

$\frac{\sqrt{1 + t}}{t (1 + t)} - \frac{1 + t}{t (1 + t)}$

Combine the numerators over the common denominator.

$\frac{\sqrt{1 + t} - (1 + t)}{t (1 + t)}$

Step 4

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + t}$ as ${(1 + t)}^{\frac{1}{2}}$ .

$\frac{{(1 + t)}^{\frac{1}{2}} - (1 + t)}{t (1 + t)}$

Rewrite $1 + t$ as ${({(1 + t)}^{\frac{1}{2}})}^{2}$ .

$\frac{{(1 + t)}^{\frac{1}{2}} - {({(1 + t)}^{\frac{1}{2}})}^{2}}{t (1 + t)}$

Let $u = {(1 + t)}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${(1 + t)}^{\frac{1}{2}}$ .

$\frac{u - u^{2}}{t (1 + t)}$

Factor $u$ out of $u - u^{2}$ .

Raise $u$ to the power of $1$ .

$\frac{u^{1} - u^{2}}{t (1 + t)}$

Factor $u$ out of $u^{1}$ .

$\frac{u \cdot 1 - u^{2}}{t (1 + t)}$

Factor $u$ out of $- u^{2}$ .

$\frac{u \cdot 1 + u (- u)}{t (1 + t)}$

Factor $u$ out of $u \cdot 1 + u (- u)$ .

$\frac{u (1 - u)}{t (1 + t)}$

Replace all occurrences of $u$ with ${(1 + t)}^{\frac{1}{2}}$ .

$\frac{{(1 + t)}^{\frac{1}{2}} (1 - {(1 + t)}^{\frac{1}{2}})}{t (1 + t)}$

Step 5

Move ${(1 + t)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1 - {(1 + t)}^{\frac{1}{2}}}{t (1 + t) {(1 + t)}^{- \frac{1}{2}}}$

Step 6

Multiply $1 + t$ by ${(1 + t)}^{- \frac{1}{2}}$ by adding the exponents.

Move ${(1 + t)}^{- \frac{1}{2}}$ .

$\frac{1 - {(1 + t)}^{\frac{1}{2}}}{t ({(1 + t)}^{- \frac{1}{2}} (1 + t))}$

Multiply ${(1 + t)}^{- \frac{1}{2}}$ by $1 + t$ .

Raise $1 + t$ to the power of $1$ .

$\frac{1 - {(1 + t)}^{\frac{1}{2}}}{t ({(1 + t)}^{- \frac{1}{2}} {(1 + t)}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1 - {(1 + t)}^{\frac{1}{2}}}{t {(1 + t)}^{- \frac{1}{2} + 1}}$

Write $1$ as a fraction with a common denominator.

$\frac{1 - {(1 + t)}^{\frac{1}{2}}}{t {(1 + t)}^{- \frac{1}{2} + \frac{2}{2}}}$

Combine the numerators over the common denominator.

$\frac{1 - {(1 + t)}^{\frac{1}{2}}}{t {(1 + t)}^{\frac{- 1 + 2}{2}}}$

Add $- 1$ and $2$ .

$\frac{1 - {(1 + t)}^{\frac{1}{2}}}{t {(1 + t)}^{\frac{1}{2}}}$

Simplify 1/(t square root of 1+t)-1/t

Related Questions

Support LCMGCF.com by disabling your adblocker.