Algebra Examples

$2 \cdot 16^{x} = 4 \cdot 8^{3 x + 2}$

Step 1

Rewrite $16$ as $2^{4}$ .

$2 \cdot {(2^{4})}^{x} = 4 \cdot 8^{3 x + 2}$

Step 2

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

$2 \cdot 2^{4 x} = 4 \cdot 8^{3 x + 2}$

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 4 x} = 4 \cdot 8^{3 x + 2}$

Step 4

Rewrite $4$ as $2^{2}$ .

$2^{1 + 4 x} = 2^{2} \cdot 8^{3 x + 2}$

Step 5

Rewrite $8$ as $2^{3}$ .

$2^{1 + 4 x} = 2^{2} \cdot {(2^{3})}^{3 x + 2}$

Step 6

Multiply the exponents in ${(2^{3})}^{3 x + 2}$ .

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Step 6.1

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

$2^{1 + 4 x} = 2^{2} \cdot 2^{3 (3 x + 2)}$

Step 6.2

Apply the distributive property.

$2^{1 + 4 x} = 2^{2} \cdot 2^{3 (3 x) + 3 \cdot 2}$

Step 6.3

Multiply $3$ by $3$ .

$2^{1 + 4 x} = 2^{2} \cdot 2^{9 x + 3 \cdot 2}$

Step 6.4

Multiply $3$ by $2$ .

$2^{1 + 4 x} = 2^{2} \cdot 2^{9 x + 6}$

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 4 x} = 2^{2 + 9 x + 6}$

Step 8

Add $2$ and $6$ .

$2^{1 + 4 x} = 2^{9 x + 8}$

Step 9

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

$1 + 4 x = 9 x + 8$

Step 10

Solve for $x$ .

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Step 10.1

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

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Step 10.1.1

Subtract $9 x$ from both sides of the equation.

$1 + 4 x - 9 x = 8$

Step 10.1.2

Subtract $9 x$ from $4 x$ .

$1 - 5 x = 8$

Step 10.2

Move all terms not containing $x$ to the right side of the equation.

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$- 5 x = 8 - 1$

Step 10.2.2

Subtract $1$ from $8$ .

$- 5 x = 7$

Step 10.3

Divide each term in $- 5 x = 7$ by $- 5$ and simplify.

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Step 10.3.1

Divide each term in $- 5 x = 7$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{7}{- 5}$

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Cancel the common factor of $- 5$ .

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Step 10.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{7}{- 5}$

Step 10.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{- 5}$

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{5}$

Decimal Form:

$x = - 1.4$

Mixed Number Form:

$x = - 1 \frac{2}{5}$