Solve for a ((x^2)^(3/5))/( fifth root of x^3)=x^a

$\frac{{(x^{2})}^{\frac{3}{5}}}{\sqrt[5]{x^{3}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{{(x^{2})}^{\frac{3}{5}}}{\sqrt[5]{x^{3}}}$ .

$x^{a} = \frac{{(x^{2})}^{\frac{3}{5}}}{\sqrt[5]{x^{3}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

$x^{a} = \frac{{(x^{2})}^{\frac{3}{5}}}{x^{\frac{3}{5}}}$

Step 3

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

$x^{a} = \frac{x^{2 (\frac{3}{5})}}{x^{\frac{3}{5}}}$

Step 4

Move $x^{\frac{3}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{2 (\frac{3}{5})} x^{- \frac{3}{5}}$

Step 5

Multiply $x^{2 (\frac{3}{5})}$ by $x^{- \frac{3}{5}}$ by adding the exponents.

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

$x^{a} = x^{2 (\frac{3}{5}) - \frac{3}{5}}$

Multiply $2 (\frac{3}{5})$ .

Combine $2$ and $\frac{3}{5}$ .

$x^{a} = x^{\frac{2 \cdot 3}{5} - \frac{3}{5}}$

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Subtract $3$ from $6$ .

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

Step 6

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

$a = \frac{3}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{3}{5}$

Decimal Form:

$a = 0.6$