Basic Math Examples

$e^{y^{2}} = e^{17 y} \cdot \frac{1}{e^{72}}$

Step 1

Move $e^{72}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$e^{y^{2}} = e^{17 y} \cdot e^{- 72}$

Step 2

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

$e^{y^{2}} = e^{17 y - 72}$

Step 3

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

$y^{2} = 17 y - 72$

Step 4

Solve for $y$ .

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Step 4.1

Subtract $17 y$ from both sides of the equation.

$y^{2} - 17 y = - 72$

Step 4.2

Add $72$ to both sides of the equation.

$y^{2} - 17 y + 72 = 0$

Step 4.3

Factor $y^{2} - 17 y + 72$ using the AC method.

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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 $72$ and whose sum is $- 17$ .

$- 9, - 8$

Step 4.3.2

Write the factored form using these integers.

$(y - 9) (y - 8) = 0$

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 $0$ .

$y - 9 = 0$

$y - 8 = 0$

Step 4.5

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

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Step 4.5.1

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

$y - 9 = 0$

Step 4.5.2

Add $9$ to both sides of the equation.

$y = 9$

Step 4.6

Set $y - 8$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $y - 8$ equal to $0$ .

$y - 8 = 0$

Step 4.6.2

Add $8$ to both sides of the equation.

$y = 8$

Step 4.7

The final solution is all the values that make $(y - 9) (y - 8) = 0$ true.

$y = 9, 8$