Algebra Examples

$2^{3} \cdot 4^{n} = 2^{11}$

Step 1

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

$2^{3} \cdot {(2^{2})}^{n} = 2^{11}$

Step 2

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

$2^{3} \cdot 2^{2 n} = 2^{11}$

Step 3

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

$2^{3 + 2 n} = 2^{11}$

Step 4

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

$3 + 2 n = 11$

Step 5

Solve for $n$ .

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Step 5.1

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

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Step 5.1.1

Subtract $3$ from both sides of the equation.

$2 n = 11 - 3$

Step 5.1.2

Subtract $3$ from $11$ .

$2 n = 8$

Step 5.2

Divide each term in $2 n = 8$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 n = 8$ by $2$ .

$\frac{2 n}{2} = \frac{8}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 n}{2} = \frac{8}{2}$

Step 5.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{8}{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $8$ by $2$ .

$n = 4$