Solve for x (4x/2)(4x/5)=4^14

$(4 \frac{x}{2}) (4 \frac{x}{5}) = 4^{14}$

Step 1

Simplify.

Combine $4$ and $\frac{x}{2}$ .

$\frac{4 x}{2} \cdot (4 (\frac{x}{5})) = 4^{14}$

Combine $4$ and $\frac{x}{5}$ .

$\frac{4 x}{2} \cdot \frac{4 x}{5} = 4^{14}$

Multiply $\frac{4 x}{2}$ by $\frac{4 x}{5}$ .

$\frac{4 x (4 x)}{2 \cdot 5} = 4^{14}$

Multiply $4$ by $4$ .

$\frac{16 x \cdot x}{2 \cdot 5} = 4^{14}$

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

$\frac{16 (x^{1} x)}{2 \cdot 5} = 4^{14}$

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

$\frac{16 (x^{1} x^{1})}{2 \cdot 5} = 4^{14}$

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

$\frac{16 x^{1 + 1}}{2 \cdot 5} = 4^{14}$

Add $1$ and $1$ .

$\frac{16 x^{2}}{2 \cdot 5} = 4^{14}$

Multiply $2$ by $5$ .

$\frac{16 x^{2}}{10} = 4^{14}$

Step 2

Divide each term in $\frac{16 x^{2}}{10} = 4^{14}$ by $16$ and simplify.

Divide each term in $\frac{16 x^{2}}{10} = 4^{14}$ by $16$ .

$\frac{\frac{16 x^{2}}{10}}{16} = \frac{4^{14}}{16}$

Simplify the left side.

Cancel the common factor of $16$ and $10$ .

Factor $2$ out of $16 x^{2}$ .

$\frac{\frac{2 (8 x^{2})}{10}}{16} = \frac{4^{14}}{16}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{\frac{2 (8 x^{2})}{2 (5)}}{16} = \frac{4^{14}}{16}$

Cancel the common factor.

$\frac{\frac{2 (8 x^{2})}{2 \cdot 5}}{16} = \frac{4^{14}}{16}$

Rewrite the expression.

$\frac{\frac{8 x^{2}}{5}}{16} = \frac{4^{14}}{16}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 x^{2}}{5} \cdot \frac{1}{16} = \frac{4^{14}}{16}$

Combine.

$\frac{8 x^{2} \cdot 1}{5 \cdot 16} = \frac{4^{14}}{16}$

Cancel the common factor of $8$ and $16$ .

Factor $8$ out of $8 x^{2} \cdot 1$ .

$\frac{8 (x^{2} \cdot 1)}{5 \cdot 16} = \frac{4^{14}}{16}$

Cancel the common factors.

Factor $8$ out of $5 \cdot 16$ .

$\frac{8 (x^{2} \cdot 1)}{8 (5 \cdot 2)} = \frac{4^{14}}{16}$

Cancel the common factor.

$\frac{8 (x^{2} \cdot 1)}{8 (5 \cdot 2)} = \frac{4^{14}}{16}$

Rewrite the expression.

$\frac{x^{2} \cdot 1}{5 \cdot 2} = \frac{4^{14}}{16}$

Multiply.

Multiply $x^{2}$ by $1$ .

$\frac{x^{2}}{5 \cdot 2} = \frac{4^{14}}{16}$

Multiply $5$ by $2$ .

$\frac{x^{2}}{10} = \frac{4^{14}}{16}$

Simplify the right side.

Raise $4$ to the power of $14$ .

$\frac{x^{2}}{10} = \frac{268435456}{16}$

Divide $268435456$ by $16$ .

$\frac{x^{2}}{10} = 16777216$

Step 3

Multiply both sides of the equation by $10$ .

$10 \frac{x^{2}}{10} = 10 \cdot 16777216$

Step 4

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$10 \frac{x^{2}}{10} = 10 \cdot 16777216$

Rewrite the expression.

$x^{2} = 10 \cdot 16777216$

Simplify the right side.

Multiply $10$ by $16777216$ .

$x^{2} = 167772160$

Step 5

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{167772160}$

Step 6

Simplify $\pm \sqrt{167772160}$ .

Rewrite $167772160$ as $4096^{2} \cdot 10$ .

Factor $16777216$ out of $167772160$ .

$x = \pm \sqrt{16777216 (10)}$

Rewrite $16777216$ as $4096^{2}$ .

$x = \pm \sqrt{4096^{2} \cdot 10}$

Pull terms out from under the radical.

$x = \pm 4096 \sqrt{10}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 4096 \sqrt{10}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4096 \sqrt{10}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4096 \sqrt{10}, - 4096 \sqrt{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 4096 \sqrt{10}, - 4096 \sqrt{10}$

Decimal Form:

$x = 12952.68929604 \dots, - 12952.68929604 \dots$

Solve for x (4x/2)(4x/5)=4^14

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