Algebra Examples

$128 \cdot 2^{\frac{x}{2}} = 64$

Step 1

Rewrite $128$ as $2^{7}$ .

$2^{7} \cdot 2^{\frac{x}{2}} = 64$

Step 2

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

$2^{7 + \frac{x}{2}} = 64$

Step 3

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

$2^{7 + \frac{x}{2}} = 2^{6}$

Step 4

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

$7 + \frac{x}{2} = 6$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $7$ from both sides of the equation.

$\frac{x}{2} = 6 - 7$

Step 5.1.2

Subtract $7$ from $6$ .

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

Step 5.2

Multiply both sides of the equation by $2$ .

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

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $2$ .

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Step 5.3.1.1.1

Cancel the common factor.

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

Step 5.3.1.1.2

Rewrite the expression.

$x = 2 \cdot - 1$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $2$ by $- 1$ .

$x = - 2$