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

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

Step 1

Rewrite the equation as $x^{a} = \frac{x^{\frac{3}{2}}}{x^{\frac{2}{5}}}$ .

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

Step 2

Move $x^{\frac{2}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 3

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

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

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

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

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

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

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

Multiply $\frac{3}{2}$ by $\frac{5}{5}$ .

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

Multiply $2$ by $5$ .

$x^{a} = x^{\frac{3 \cdot 5}{10} - \frac{2}{5} \cdot \frac{2}{2}}$

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

$x^{a} = x^{\frac{3 \cdot 5}{10} - \frac{2 \cdot 2}{5 \cdot 2}}$

Multiply $5$ by $2$ .

$x^{a} = x^{\frac{3 \cdot 5}{10} - \frac{2 \cdot 2}{10}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{3 \cdot 5 - 2 \cdot 2}{10}}$

Simplify the numerator.

Multiply $3$ by $5$ .

$x^{a} = x^{\frac{15 - 2 \cdot 2}{10}}$

Multiply $- 2$ by $2$ .

$x^{a} = x^{\frac{15 - 4}{10}}$

Subtract $4$ from $15$ .

$x^{a} = x^{\frac{11}{10}}$

Step 4

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

$a = \frac{11}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{11}{10}$

Decimal Form:

$a = 1.1$

Mixed Number Form:

$a = 1 \frac{1}{10}$