Precalculus Examples

$e^{x^{2}} = e^{13 x} \cdot \frac{1}{e^{36}}$

Step 1

Move $e^{36}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$e^{x^{2}} = e^{13 x} \cdot e^{- 36}$

Step 2

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

$e^{x^{2}} = e^{13 x - 36}$

Step 3

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

$x^{2} = 13 x - 36$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Subtract $13 x$ from both sides of the equation.

$x^{2} - 13 x = - 36$

Step 4.2

Add $36$ to both sides of the equation.

$x^{2} - 13 x + 36 = 0$

Step 4.3

Factor $x^{2} - 13 x + 36$ using the AC method.

Tap for more steps...

Step 4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $36$ and whose sum is $- 13$ .

$- 9, - 4$

Step 4.3.2

Write the factored form using these integers.

$(x - 9) (x - 4) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 9 = 0$

$x - 4 = 0$

Step 4.5

Set $x - 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.5.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 4.5.2

Add $9$ to both sides of the equation.

$x = 9$

Step 4.6

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.7

The final solution is all the values that make $(x - 9) (x - 4) = 0$ true.

$x = 9, 4$