Calculus Examples

$2 e^{4 x} = e^{4 x} + e^{8}$

Step 1

Rewrite $e^{4 x}$ as exponentiation.

$2 e^{4 x} = {(e^{x})}^{4} + e^{8} = e^{4 x} + e^{8}$

Step 2

Substitute $u$ for $e^{x}$ .

$2 e^{4 x} = u^{4} + e^{8} = e^{4 x} + e^{8}$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $e^{4 x}$ from both sides of the equation.

$2 e^{4 x} - e^{4 x} = e^{8}$

Step 3.2

Subtract $e^{4 x}$ from $2 e^{4 x}$ .

$e^{4 x} = e^{8}$

Step 4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$4 x = 8$

Step 5

Divide each term in $4 x = 8$ by $4$ and simplify.

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Step 5.1

Divide each term in $4 x = 8$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $8$ by $4$ .

$x = 2$