Precalculus Examples

$2^{x} + 16 (2^{- x}) = 10$

Step 1

Rewrite $2^{- x}$ as exponentiation.

$2^{x} + 16 \cdot {(2^{x})}^{- 1} = 10$

Step 2

Substitute $u$ for $2^{x}$ .

$u + 16 \cdot u^{- 1} = 10$

Step 3

Simplify each term.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$u + 16 \cdot \frac{1}{u} = 10$

Step 3.2

Combine $16$ and $\frac{1}{u}$ .

$u + \frac{16}{u} = 10$

Step 4

Solve for $u$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, u, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$u$

Step 4.2

Multiply each term in $u + \frac{16}{u} = 10$ by $u$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $u + \frac{16}{u} = 10$ by $u$ .

$u \cdot u + \frac{16}{u} u = 10 u$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} + \frac{16}{u} u = 10 u$

Step 4.2.2.1.2

Cancel the common factor of $u$ .

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Step 4.2.2.1.2.1

Cancel the common factor.

$u^{2} + \frac{16}{u} u = 10 u$

Step 4.2.2.1.2.2

Rewrite the expression.

$u^{2} + 16 = 10 u$

Step 4.3

Solve the equation.

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Step 4.3.1

Subtract $10 u$ from both sides of the equation.

$u^{2} + 16 - 10 u = 0$

Step 4.3.2

Factor $u^{2} + 16 - 10 u$ using the AC method.

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Step 4.3.2.1

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 $16$ and whose sum is $- 10$ .

$- 8, - 2$

Step 4.3.2.2

Write the factored form using these integers.

$(u - 8) (u - 2) = 0$

Step 4.3.3

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

$u - 8 = 0$

$u - 2 = 0$

Step 4.3.4

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

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Step 4.3.4.1

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

$u - 8 = 0$

Step 4.3.4.2

Add $8$ to both sides of the equation.

$u = 8$

Step 4.3.5

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 4.3.5.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 4.3.5.2

Add $2$ to both sides of the equation.

$u = 2$

Step 4.3.6

The final solution is all the values that make $(u - 8) (u - 2) = 0$ true.

$u = 8, 2$

Step 5

Substitute $8$ for $u$ in $u = 2^{x}$ .

$8 = 2^{x}$

Step 6

Solve $8 = 2^{x}$ .

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Step 6.1

Rewrite the equation as $2^{x} = 8$ .

$2^{x} = 8$

Step 6.2

Create equivalent expressions in the equation that all have equal bases.

$2^{x} = 2^{3}$

Step 6.3

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

$x = 3$

Step 7

Substitute $2$ for $u$ in $u = 2^{x}$ .

$2 = 2^{x}$

Step 8

Solve $2 = 2^{x}$ .

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Step 8.1

Rewrite the equation as $2^{x} = 2$ .

$2^{x} = 2$

Step 8.2

Create equivalent expressions in the equation that all have equal bases.

$2^{x} = 2^{1}$

Step 8.3

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

$x = 1$

Step 9

List the solutions that makes the equation true.

$x = 3, 1$