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

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

Step 1

Multiply $4^{\frac{x}{2}}$ by $4^{\frac{x}{5}}$ by adding the exponents.

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

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

To write $\frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

To write $\frac{x}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $2$ by $5$ .

$4^{\frac{x \cdot 5}{10} + \frac{x}{5} \cdot \frac{2}{2}} = 4^{14}$

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

$4^{\frac{x \cdot 5}{10} + \frac{x \cdot 2}{5 \cdot 2}} = 4^{14}$

Multiply $5$ by $2$ .

$4^{\frac{x \cdot 5}{10} + \frac{x \cdot 2}{10}} = 4^{14}$

Combine the numerators over the common denominator.

$4^{\frac{x \cdot 5 + x \cdot 2}{10}} = 4^{14}$

Simplify the numerator.

Move $5$ to the left of $x$ .

$4^{\frac{5 \cdot x + x \cdot 2}{10}} = 4^{14}$

Move $2$ to the left of $x$ .

$4^{\frac{5 x + 2 \cdot x}{10}} = 4^{14}$

Add $5 x$ and $2 x$ .

$4^{\frac{7 x}{10}} = 4^{14}$

Step 2

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

$\frac{7 x}{10} = 14$

Step 3

Solve for $x$ .

Multiply both sides of the equation by $\frac{10}{7}$ .

$\frac{10}{7} \cdot \frac{7 x}{10} = \frac{10}{7} \cdot 14$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{10}{7} \cdot \frac{7 x}{10}$ .

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10}{7} \cdot \frac{7 x}{10} = \frac{10}{7} \cdot 14$

Rewrite the expression.

$\frac{1}{7} (7 x) = \frac{10}{7} \cdot 14$

Cancel the common factor of $7$ .

Factor $7$ out of $7 x$ .

$\frac{1}{7} (7 (x)) = \frac{10}{7} \cdot 14$

Cancel the common factor.

$\frac{1}{7} (7 x) = \frac{10}{7} \cdot 14$

Rewrite the expression.

$x = \frac{10}{7} \cdot 14$

Simplify the right side.

Simplify $\frac{10}{7} \cdot 14$ .

Cancel the common factor of $7$ .

Factor $7$ out of $14$ .

$x = \frac{10}{7} \cdot (7 (2))$

Cancel the common factor.

$x = \frac{10}{7} \cdot (7 \cdot 2)$

Rewrite the expression.

$x = 10 \cdot 2$

Multiply $10$ by $2$ .

$x = 20$

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

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