Algebra Examples

${(3^{x})}^{x + 2} = \frac{1}{3}$

Step 1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{x (x + 2)} = \frac{1}{3}$

Step 2

Raise $3$ to the power of $1$ .

$3^{x (x + 2)} = \frac{1}{3^{1}}$

Step 3

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{x (x + 2)} = 3^{- 1}$

Step 4

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

$x (x + 2) = - 1$

Step 5

Solve for $x$ .

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Step 5.1

Simplify $x (x + 2)$ .

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Step 5.1.1

Apply the distributive property.

$x \cdot x + x \cdot 2 = - 1$

Step 5.1.2

Simplify the expression.

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Step 5.1.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 = - 1$

Step 5.1.2.2

Move $2$ to the left of $x$ .

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

Step 5.2

Add $1$ to both sides of the equation.

$x^{2} + 2 x + 1 = 0$

Step 5.3

Factor using the perfect square rule.

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Step 5.3.1

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

$x^{2} + 2 x + 1^{2} = 0$

Step 5.3.2

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

$2 x = 2 \cdot x \cdot 1$

Step 5.3.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 1 + 1^{2} = 0$

Step 5.3.4

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

${(x + 1)}^{2} = 0$

Step 5.4

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 5.5

Subtract $1$ from both sides of the equation.

$x = - 1$