Basic Math Examples

$3^{1 - a} - 3^{a} = 2$

Step 1

Rewrite $3^{1 - a}$ as $3^{1} \cdot 3^{- a}$ .

$3 \cdot 3^{- a} - 3^{a} = 2$

Step 2

Rewrite $3^{- a}$ as exponentiation.

$3 \cdot {(3^{a})}^{- 1} - 3^{a} = 2$

Step 3

Substitute $u$ for $3^{a}$ .

$3 \cdot u^{- 1} - u = 2$

Step 4

Simplify each term.

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Step 4.1

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

$3 \cdot \frac{1}{u} - u = 2$

Step 4.2

Combine $3$ and $\frac{1}{u}$ .

$\frac{3}{u} - u = 2$

Step 5

Reorder $\frac{3}{u}$ and $- u$ .

$- u + \frac{3}{u} = 2$

Step 6

Solve for $u$ .

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Step 6.1

Find the LCD of the terms in the equation.

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Step 6.1.1

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 6.1.2

The LCM of one and any expression is the expression.

$u$

Step 6.2

Multiply each term in $- u + \frac{3}{u} = 2$ by $u$ to eliminate the fractions.

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Step 6.2.1

Multiply each term in $- u + \frac{3}{u} = 2$ by $u$ .

$- u \cdot u + \frac{3}{u} u = 2 u$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $u$ by $u$ by adding the exponents.

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Step 6.2.2.1.1.1

Move $u$ .

$- (u \cdot u) + \frac{3}{u} u = 2 u$

Step 6.2.2.1.1.2

Multiply $u$ by $u$ .

$- u^{2} + \frac{3}{u} u = 2 u$

Step 6.2.2.1.2

Cancel the common factor of $u$ .

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Step 6.2.2.1.2.1

Cancel the common factor.

$- u^{2} + \frac{3}{u} u = 2 u$

Step 6.2.2.1.2.2

Rewrite the expression.

$- u^{2} + 3 = 2 u$

Step 6.3

Solve the equation.

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Step 6.3.1

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

$- u^{2} + 3 - 2 u = 0$

Step 6.3.2

Factor the left side of the equation.

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Step 6.3.2.1

Factor $- 1$ out of $- u^{2} + 3 - 2 u$ .

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Step 6.3.2.1.1

Move $3$ .

$- u^{2} - 2 u + 3 = 0$

Step 6.3.2.1.2

Factor $- 1$ out of $- u^{2}$ .

$- (u^{2}) - 2 u + 3 = 0$

Step 6.3.2.1.3

Factor $- 1$ out of $- 2 u$ .

$- (u^{2}) - (2 u) + 3 = 0$

Step 6.3.2.1.4

Rewrite $3$ as $- 1 (- 3)$ .

$- (u^{2}) - (2 u) - 1 \cdot - 3 = 0$

Step 6.3.2.1.5

Factor $- 1$ out of $- (u^{2}) - (2 u)$ .

$- (u^{2} + 2 u) - 1 \cdot - 3 = 0$

Step 6.3.2.1.6

Factor $- 1$ out of $- (u^{2} + 2 u) - 1 (- 3)$ .

$- (u^{2} + 2 u - 3) = 0$

Step 6.3.2.2

Factor.

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Step 6.3.2.2.1

Factor $u^{2} + 2 u - 3$ using the AC method.

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Step 6.3.2.2.1.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 6.3.2.2.1.2

Write the factored form using these integers.

$- ((u - 1) (u + 3)) = 0$

Step 6.3.2.2.2

Remove unnecessary parentheses.

$- (u - 1) (u + 3) = 0$

Step 6.3.3

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

$u - 1 = 0$

$u + 3 = 0$

Step 6.3.4

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 6.3.4.1

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

$u - 1 = 0$

Step 6.3.4.2

Add $1$ to both sides of the equation.

$u = 1$

Step 6.3.5

Set $u + 3$ equal to $0$ and solve for $u$ .

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Step 6.3.5.1

Set $u + 3$ equal to $0$ .

$u + 3 = 0$

Step 6.3.5.2

Subtract $3$ from both sides of the equation.

$u = - 3$

Step 6.3.6

The final solution is all the values that make $- (u - 1) (u + 3) = 0$ true.

$u = 1, - 3$

Step 7

Substitute $1$ for $u$ in $u = 3^{a}$ .

$1 = 3^{a}$

Step 8

Solve $1 = 3^{a}$ .

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Step 8.1

Rewrite the equation as $3^{a} = 1$ .

$3^{a} = 1$

Step 8.2

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

$\ln (3^{a}) = \ln (1)$

Step 8.3

Expand $\ln (3^{a})$ by moving $a$ outside the logarithm.

$a \ln (3) = \ln (1)$

Step 8.4

Simplify the right side.

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Step 8.4.1

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

$a \ln (3) = 0$

Step 8.5

Divide each term in $a \ln (3) = 0$ by $\ln (3)$ and simplify.

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Step 8.5.1

Divide each term in $a \ln (3) = 0$ by $\ln (3)$ .

$\frac{a \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Step 8.5.2

Simplify the left side.

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Step 8.5.2.1

Cancel the common factor of $\ln (3)$ .

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Step 8.5.2.1.1

Cancel the common factor.

$\frac{a \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Step 8.5.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{\ln (3)}$

Step 8.5.3

Simplify the right side.

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Step 8.5.3.1

Divide $0$ by $\ln (3)$ .

$a = 0$

Step 9

Substitute $- 3$ for $u$ in $u = 3^{a}$ .

$- 3 = 3^{a}$

Step 10

Solve $- 3 = 3^{a}$ .

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Step 10.1

Rewrite the equation as $3^{a} = - 3$ .

$3^{a} = - 3$

Step 10.2

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

$\ln (3^{a}) = \ln (- 3)$

Step 10.3

The equation cannot be solved because $\ln (- 3)$ is undefined.

Undefined

Step 10.4

There is no solution for $3^{a} = - 3$

No solution

Step 11

List the solutions that makes the equation true.

$a = 0$