Solve for x 3=(729^(1/5))(3^(x/5))

$3 = (729^{\frac{1}{5}}) (3^{\frac{x}{5}})$

Step 1

Rewrite the equation as $729^{\frac{1}{5}} \cdot 3^{\frac{x}{5}} = 3$ .

$729^{\frac{1}{5}} \cdot 3^{\frac{x}{5}} = 3$

Step 2

Rewrite $729$ as $3^{6}$ .

${(3^{6})}^{\frac{1}{5}} \cdot 3^{\frac{x}{5}} = 3$

Step 3

Multiply the exponents in ${(3^{6})}^{\frac{1}{5}}$ .

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

$3^{6 (\frac{1}{5})} \cdot 3^{\frac{x}{5}} = 3$

Combine $6$ and $\frac{1}{5}$ .

$3^{\frac{6}{5}} \cdot 3^{\frac{x}{5}} = 3$

Step 4

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

$3^{\frac{6}{5} + \frac{x}{5}} = 3$

Step 5

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

$3^{\frac{6}{5} + \frac{x}{5}} = 3^{1}$

Step 6

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

$\frac{6}{5} + \frac{x}{5} = 1$

Step 7

Solve for $x$ .

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

Subtract $\frac{6}{5}$ from both sides of the equation.

$\frac{x}{5} = 1 - \frac{6}{5}$

Write $1$ as a fraction with a common denominator.

$\frac{x}{5} = \frac{5}{5} - \frac{6}{5}$

Combine the numerators over the common denominator.

$\frac{x}{5} = \frac{5 - 6}{5}$

Subtract $6$ from $5$ .

$\frac{x}{5} = \frac{- 1}{5}$

Move the negative in front of the fraction.

$\frac{x}{5} = - \frac{1}{5}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - 1$