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

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

Step 1

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

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

Step 2

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

Multiply $x$ by $x^{\frac{1}{2}}$ .

Raise $x$ to the power of $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Step 3

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

$a = \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$a = \frac{3}{2}$

Decimal Form:

$a = 1.5$

Mixed Number Form:

$a = 1 \frac{1}{2}$