Solve for a x^(2/3)x^(1/2)=x^a

$x^{\frac{2}{3}} x^{\frac{1}{2}} = x^{a}$

Rewrite the equation as $x^{a} = x^{\frac{2}{3}} x^{\frac{1}{2}}$ .

$x^{a} = x^{\frac{2}{3}} x^{\frac{1}{2}}$

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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

$x^{a} = x^{\frac{2}{3} + \frac{1}{2}}$

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

$x^{a} = x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2}}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{a} = x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$x^{a} = x^{\frac{2 \cdot 2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}}$

Multiply $3$ by $2$ .

$x^{a} = x^{\frac{2 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3}}$

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$x^{a} = x^{\frac{2 \cdot 2}{6} + \frac{3}{2 \cdot 3}}$

Multiply $2$ by $3$ .

$x^{a} = x^{\frac{2 \cdot 2}{6} + \frac{3}{6}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{2 \cdot 2 + 3}{6}}$

Simplify the numerator.

Multiply $2$ by $2$ .

$x^{a} = x^{\frac{4 + 3}{6}}$

Add $4$ and $3$ .

$x^{a} = x^{\frac{7}{6}}$

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

$a = \frac{7}{6}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{7}{6}$

Decimal Form:

$a = 1.1\overline{6}$

Mixed Number Form:

$a = 1 \frac{1}{6}$