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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 10.30 (page 1015)

Solve:
$\frac{e^{x^{2}}}{e^{x}} = 6$

Short Answer

Expert verified

By solving $\frac{e^{x^{2}}}{e^{x}} = 6$ , the value of $x$ is $- 2, 3$

Step by step solution

01

Step 1. Given

An expression $\frac{e^{x^{2}}}{e^{x}} = 6$

To solve the expression.

02

Step 2. Use the property of exponent

By using the property of exponent, $\frac{a^{m}}{a^{n}} = a^{m - n}$ ,

$e^{x^{2} - x} = 6$

03

Step 3. Write a new equation by setting the exponents equal.

Since the bases are same, the exponents must be equal.

$x^{2} - x = 6 x^{2} - x - 6 = 0 x^{2} - 3 x + 2 x - 6 = 0 (x - 3) (x + 2) = 0 x = - 2, 3$

The value of $x$ is $- 2, 3$

04

Step 4. Check the solutions.

When $x = - 2$ ,

$\frac{e^{{(- 2)}^{2}}}{e^{- 2}} = e^{6} \frac{e^{4}}{e^{- 2}} = e^{6} e^{4 + 2} = e^{6} e^{6} = e^{6}$

When $x = 3$ ,

$\frac{e^{3^{2}}}{e^{3}} = e^{6} e^{9 - 3} = e^{6} e^{6} = e^{6}$

Hence the solution has been checked.

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