Algebra Examples

$2 (4^{2 x + 1}) = 128$

Step 1

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

$2 \cdot {(2^{2})}^{2 x + 1} = 128$

Step 2

Multiply the exponents in ${(2^{2})}^{2 x + 1}$ .

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Step 2.1

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

$2 \cdot 2^{2 (2 x + 1)} = 128$

Step 2.2

Apply the distributive property.

$2 \cdot 2^{2 (2 x) + 2 \cdot 1} = 128$

Step 2.3

Multiply $2$ by $2$ .

$2 \cdot 2^{4 x + 2 \cdot 1} = 128$

Step 2.4

Multiply $2$ by $1$ .

$2 \cdot 2^{4 x + 2} = 128$

Step 3

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

$2^{1 + 4 x + 2} = 128$

Step 4

Add $1$ and $2$ .

$2^{4 x + 3} = 128$

Step 5

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

$2^{4 x + 3} = 2^{7}$

Step 6

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

$4 x + 3 = 7$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Subtract $3$ from both sides of the equation.

$4 x = 7 - 3$

Step 7.1.2

Subtract $3$ from $7$ .

$4 x = 4$

Step 7.2

Divide each term in $4 x = 4$ by $4$ and simplify.

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Step 7.2.1

Divide each term in $4 x = 4$ by $4$ .

$\frac{4 x}{4} = \frac{4}{4}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{4}{4}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{4}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $4$ by $4$ .

$x = 1$