Algebra Examples

$2^{x} + 2^{- x} = 2$

Step 1

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

$2^{x} + {(2^{x})}^{- 1} = 2$

Step 2

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

$u + u^{- 1} = 2$

Step 3

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

$u + \frac{1}{u} = 2$

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 $u + \frac{1}{u} = 2$ by $u$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $u + \frac{1}{u} = 2$ by $u$ .

$u \cdot u + \frac{1}{u} u = 2 u$

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 $u$ .

$u^{2} + \frac{1}{u} u = 2 u$

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.

$u^{2} + \frac{1}{u} u = 2 u$

Step 4.2.2.1.2.2

Rewrite the expression.

$u^{2} + 1 = 2 u$

Step 4.3

Solve the equation.

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Step 4.3.1

Subtract $2 u$ from both sides of the equation.

$u^{2} + 1 - 2 u = 0$

Step 4.3.2

Factor using the perfect square rule.

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Step 4.3.2.1

Rearrange terms.

$u^{2} - 2 u + 1 = 0$

Step 4.3.2.2

Rewrite $1$ as $1^{2}$ .

$u^{2} - 2 u + 1^{2} = 0$

Step 4.3.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 4.3.2.4

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 1 + 1^{2} = 0$

Step 4.3.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 1$ .

${(u - 1)}^{2} = 0$

Step 4.3.3

Set the $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 4.3.4

Add $1$ to both sides of the equation.

$u = 1$

Step 5

Substitute $1$ for $u$ in $u = 2^{x}$ .

$1 = 2^{x}$

Step 6

Solve $1 = 2^{x}$ .

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Step 6.1

Rewrite the equation as $2^{x} = 1$ .

$2^{x} = 1$

Step 6.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (2^{x}) = \ln (1)$

Step 6.3

Expand $\ln (2^{x})$ by moving $x$ outside the logarithm.

$x \ln (2) = \ln (1)$

Step 6.4

Simplify the right side.

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Step 6.4.1

The natural logarithm of $1$ is $0$ .

$x \ln (2) = 0$

Step 6.5

Divide each term in $x \ln (2) = 0$ by $\ln (2)$ and simplify.

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Step 6.5.1

Divide each term in $x \ln (2) = 0$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{0}{\ln (2)}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $\ln (2)$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{0}{\ln (2)}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{\ln (2)}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $0$ by $\ln (2)$ .

$x = 0$