Solve for a (x^(1/2))/( cube root of x^5)=x^a

$\frac{x^{\frac{1}{2}}}{\sqrt[3]{x^{5}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{x^{\frac{1}{2}}}{\sqrt[3]{x^{5}}}$ .

$x^{a} = \frac{x^{\frac{1}{2}}}{\sqrt[3]{x^{5}}}$

Step 2

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

$x^{a} = \frac{x^{\frac{1}{2}}}{x^{\frac{5}{3}}}$

Step 3

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

$x^{a} = x^{\frac{1}{2}} x^{- \frac{5}{3}}$

Step 4

Multiply $x^{\frac{1}{2}}$ by $x^{- \frac{5}{3}}$ by adding the exponents.

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

$x^{a} = x^{\frac{1}{2} - \frac{5}{3}}$

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

$x^{a} = x^{\frac{1}{2} \cdot \frac{3}{3} - \frac{5}{3}}$

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

$x^{a} = x^{\frac{1}{2} \cdot \frac{3}{3} - \frac{5}{3} \cdot \frac{2}{2}}$

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

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

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

Multiply $2$ by $3$ .

$x^{a} = x^{\frac{3}{6} - \frac{5}{3} \cdot \frac{2}{2}}$

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

$x^{a} = x^{\frac{3}{6} - \frac{5 \cdot 2}{3 \cdot 2}}$

Multiply $3$ by $2$ .

$x^{a} = x^{\frac{3}{6} - \frac{5 \cdot 2}{6}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{3 - 5 \cdot 2}{6}}$

Simplify the numerator.

Multiply $- 5$ by $2$ .

$x^{a} = x^{\frac{3 - 10}{6}}$

Subtract $10$ from $3$ .

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

Move the negative in front of the fraction.

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

Step 5

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

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

Step 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:

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

Solve for a (x^(1/2))/( cube root of x^5)=x^a

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