Simplify 2e^(3 natural log of x+1)

$2 e^{3 \ln (x + 1)}$

Step 1

Simplify $3 \ln (x + 1)$ by moving $3$ inside the logarithm.

$2 e^{\ln ({(x + 1)}^{3})}$

Step 2

Exponentiation and log are inverse functions.

$2 {(x + 1)}^{3}$

Step 3

Use the Binomial Theorem.

$2 (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3})$

Step 4

Simplify terms.

Simplify each term.

Multiply $3$ by $1$ .

$2 (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3})$

One to any power is one.

$2 (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3})$

Multiply $3$ by $1$ .

$2 (x^{3} + 3 x^{2} + 3 x + 1^{3})$

One to any power is one.

$2 (x^{3} + 3 x^{2} + 3 x + 1)$

Apply the distributive property.

$2 x^{3} + 2 (3 x^{2}) + 2 (3 x) + 2 \cdot 1$

Step 5

Simplify.

Multiply $3$ by $2$ .

$2 x^{3} + 6 x^{2} + 2 (3 x) + 2 \cdot 1$

Multiply $3$ by $2$ .

$2 x^{3} + 6 x^{2} + 6 x + 2 \cdot 1$

Multiply $2$ by $1$ .

$2 x^{3} + 6 x^{2} + 6 x + 2$