Solve for x seventh root of (5^16+5^x)/(5^x+5^2)=5

$\sqrt[7]{\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}}} = 5$

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${\sqrt[7]{\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}}}}^{7} = 5^{7}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}}}$ as ${(\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}})}^{\frac{1}{7}}$ .

${({(\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}})}^{\frac{1}{7}})}^{7} = 5^{7}$

Simplify the left side.

Simplify ${({(\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}})}^{\frac{1}{7}})}^{7}$ .

Multiply the exponents in ${({(\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}})}^{\frac{1}{7}})}^{7}$ .

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

${(\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}})}^{\frac{1}{7} \cdot 7} = 5^{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

${(\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}})}^{\frac{1}{7} \cdot 7} = 5^{7}$

Rewrite the expression.

${(\frac{5^{16} + 5^{x}}{5^{x} + 5^{2}})}^{1} = 5^{7}$

Raise $5$ to the power of $16$ .

${(\frac{152587890625 + 5^{x}}{5^{x} + 5^{2}})}^{1} = 5^{7}$

Raise $5$ to the power of $2$ .

${(\frac{152587890625 + 5^{x}}{5^{x} + 25})}^{1} = 5^{7}$

Simplify.

$\frac{152587890625 + 5^{x}}{5^{x} + 25} = 5^{7}$

Simplify the right side.

Raise $5$ to the power of $7$ .

$\frac{152587890625 + 5^{x}}{5^{x} + 25} = 78125$

Solve for $x$ .

Multiply both sides by $5^{x} + 25$ .

$\frac{152587890625 + 5^{x}}{5^{x} + 25} (5^{x} + 25) = 78125 (5^{x} + 25)$

Simplify.

Simplify the left side.

Cancel the common factor of $5^{x} + 25$ .

Cancel the common factor.

$\frac{152587890625 + 5^{x}}{5^{x} + 25} (5^{x} + 25) = 78125 (5^{x} + 25)$

Rewrite the expression.

$152587890625 + 5^{x} = 78125 (5^{x} + 25)$

Simplify the right side.

Simplify $78125 (5^{x} + 25)$ .

Apply the distributive property.

$152587890625 + 5^{x} = 78125 \cdot 5^{x} + 78125 \cdot 25$

Multiply $78125 \cdot 5^{x}$ .

Rewrite $78125$ as $5^{7}$ .

$152587890625 + 5^{x} = 5^{7} \cdot 5^{x} + 78125 \cdot 25$

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

$152587890625 + 5^{x} = 5^{7 + x} + 78125 \cdot 25$

Multiply $78125$ by $25$ .

$152587890625 + 5^{x} = 5^{7 + x} + 1953125$

Reorder $7$ and $x$ .

$152587890625 + 5^{x} = 5^{x + 7} + 1953125$

Solve for $x$ .

Factor out $5^{x}$ from the expression.

$5^{x} (1 - 78125)$

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

Subtract $152587890625$ from both sides of the equation.

$5^{x} = 5^{x + 7} + 1953125 - 152587890625$

Subtract $152587890625$ from $1953125$ .

$5^{x} = 5^{x + 7} - 152585937500$

Subtract $5^{x + 7}$ from both sides of the equation.

$5^{x} - 5^{x + 7} = - 152585937500$

Subtract $78125$ from $1$ .

$5^{x} \cdot - 78124 = - 152585937500$

Move $- 78124$ to the left of $5^{x}$ .

$- 78124 \cdot 5^{x} = - 152585937500$

Divide each term in $(- 78124 \cdot 5^{x}) = - 152585937500$ by $- 78124$ and simplify.

Divide each term in $(- 78124 \cdot 5^{x}) = - 152585937500$ by $- 78124$ .

$\frac{(- 78124 \cdot 5^{x})}{- 78124} = \frac{- 152585937500}{- 78124}$

Simplify the left side.

Cancel the common factor of $- 78124$ .

Cancel the common factor.

$\frac{- 78124 \cdot 5^{x}}{- 78124} = \frac{- 152585937500}{- 78124}$

Divide $5^{x}$ by $1$ .

$5^{x} = \frac{- 152585937500}{- 78124}$

Simplify the right side.

Divide $- 152585937500$ by $- 78124$ .

$5^{x} = 1953125$

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

$5^{x} = 5^{9}$

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

$x = 9$

Solve for x seventh root of (5^16+5^x)/(5^x+5^2)=5

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