Algebra Examples

${(\frac{1}{216})}^{10 x + 16} = {(\frac{1}{6})}^{- 2 x^{2} + 8 x}$

Step 1

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

${(\frac{1}{6})}^{3 (10 x + 16)} = {(\frac{1}{6})}^{- 2 x^{2} + 8 x}$

Step 2

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

$3 (10 x + 16) = - 2 x^{2} + 8 x$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 2 x^{2} + 8 x = 3 (10 x + 16)$

Step 3.2

Simplify $3 (10 x + 16)$ .

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Step 3.2.1

Apply the distributive property.

$- 2 x^{2} + 8 x = 3 (10 x) + 3 \cdot 16$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $10$ by $3$ .

$- 2 x^{2} + 8 x = 30 x + 3 \cdot 16$

Step 3.2.2.2

Multiply $3$ by $16$ .

$- 2 x^{2} + 8 x = 30 x + 48$

Step 3.3

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.1

Subtract $30 x$ from both sides of the equation.

$- 2 x^{2} + 8 x - 30 x = 48$

Step 3.3.2

Subtract $30 x$ from $8 x$ .

$- 2 x^{2} - 22 x = 48$

Step 3.4

Subtract $48$ from both sides of the equation.

$- 2 x^{2} - 22 x - 48 = 0$

Step 3.5

Factor the left side of the equation.

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Step 3.5.1

Factor $- 2$ out of $- 2 x^{2} - 22 x - 48$ .

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Step 3.5.1.1

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} - 22 x - 48 = 0$

Step 3.5.1.2

Factor $- 2$ out of $- 22 x$ .

$- 2 x^{2} - 2 (11 x) - 48 = 0$

Step 3.5.1.3

Factor $- 2$ out of $- 48$ .

$- 2 x^{2} - 2 (11 x) - 2 \cdot 24 = 0$

Step 3.5.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (11 x)$ .

$- 2 (x^{2} + 11 x) - 2 \cdot 24 = 0$

Step 3.5.1.5

Factor $- 2$ out of $- 2 (x^{2} + 11 x) - 2 (24)$ .

$- 2 (x^{2} + 11 x + 24) = 0$

Step 3.5.2

Factor.

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Step 3.5.2.1

Factor $x^{2} + 11 x + 24$ using the AC method.

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Step 3.5.2.1.1

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 $24$ and whose sum is $11$ .

$3, 8$

Step 3.5.2.1.2

Write the factored form using these integers.

$- 2 ((x + 3) (x + 8)) = 0$

Step 3.5.2.2

Remove unnecessary parentheses.

$- 2 (x + 3) (x + 8) = 0$

Step 3.6

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

$x + 3 = 0$

$x + 8 = 0$

Step 3.7

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

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Step 3.7.1

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

$x + 3 = 0$

Step 3.7.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.8

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 3.8.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 3.9

The final solution is all the values that make $- 2 (x + 3) (x + 8) = 0$ true.

$x = - 3, - 8$