Solve for a (x^(5/3))/(x^(6/5))=x^a

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

Rewrite the equation as $x^{a} = \frac{x^{\frac{5}{3}}}{x^{\frac{6}{5}}}$ .

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

Move $x^{\frac{6}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{5}{3}} x^{- \frac{6}{5}}$

Multiply $x^{\frac{5}{3}}$ by $x^{- \frac{6}{5}}$ by adding the exponents.

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

$x^{a} = x^{\frac{5}{3} - \frac{6}{5}}$

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

To write $- \frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{3}$ by $\frac{5}{5}$ .

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

Multiply $3$ by $5$ .

$x^{a} = x^{\frac{5 \cdot 5}{15} - \frac{6}{5} \cdot \frac{3}{3}}$

Multiply $\frac{6}{5}$ by $\frac{3}{3}$ .

$x^{a} = x^{\frac{5 \cdot 5}{15} - \frac{6 \cdot 3}{5 \cdot 3}}$

Multiply $5$ by $3$ .

$x^{a} = x^{\frac{5 \cdot 5}{15} - \frac{6 \cdot 3}{15}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{5 \cdot 5 - 6 \cdot 3}{15}}$

Simplify the numerator.

Multiply $5$ by $5$ .

$x^{a} = x^{\frac{25 - 6 \cdot 3}{15}}$

Multiply $- 6$ by $3$ .

$x^{a} = x^{\frac{25 - 18}{15}}$

Subtract $18$ from $25$ .

$x^{a} = x^{\frac{7}{15}}$

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

$a = \frac{7}{15}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{7}{15}$

Decimal Form:

$a = 0.4\overline{6}$