Simplify the Radical Expression (2x^(-7/4))/(4x^(4/3))

$\frac{2 x^{- \frac{7}{4}}}{4 x^{\frac{4}{3}}}$

Move $x^{- \frac{7}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{4 x^{\frac{4}{3}} x^{\frac{7}{4}}}$

Multiply $x^{\frac{4}{3}}$ by $x^{\frac{7}{4}}$ by adding the exponents.

Move $x^{\frac{7}{4}}$ .

$\frac{2}{4 (x^{\frac{7}{4}} x^{\frac{4}{3}})}$

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

$\frac{2}{4 x^{\frac{7}{4} + \frac{4}{3}}}$

To write $\frac{7}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{4 x^{\frac{7}{4} \cdot \frac{3}{3} + \frac{4}{3}}}$

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2}{4 x^{\frac{7}{4} \cdot \frac{3}{3} + \frac{4}{3} \cdot \frac{4}{4}}}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$\frac{2}{4 x^{\frac{7 \cdot 3}{4 \cdot 3} + \frac{4}{3} \cdot \frac{4}{4}}}$

Multiply $4$ by $3$ .

$\frac{2}{4 x^{\frac{7 \cdot 3}{12} + \frac{4}{3} \cdot \frac{4}{4}}}$

Multiply $\frac{4}{3}$ by $\frac{4}{4}$ .

$\frac{2}{4 x^{\frac{7 \cdot 3}{12} + \frac{4 \cdot 4}{3 \cdot 4}}}$

Multiply $3$ by $4$ .

$\frac{2}{4 x^{\frac{7 \cdot 3}{12} + \frac{4 \cdot 4}{12}}}$

Combine the numerators over the common denominator.

$\frac{2}{4 x^{\frac{7 \cdot 3 + 4 \cdot 4}{12}}}$

Simplify the numerator.

Multiply $7$ by $3$ .

$\frac{2}{4 x^{\frac{21 + 4 \cdot 4}{12}}}$

Multiply $4$ by $4$ .

$\frac{2}{4 x^{\frac{21 + 16}{12}}}$

Add $21$ and $16$ .

$\frac{2}{4 x^{\frac{37}{12}}}$

Factor $2$ out of $2$ .

$\frac{2 (1)}{4 x^{\frac{37}{12}}}$

Cancel the common factors.

Factor $2$ out of $4 x^{\frac{37}{12}}$ .

$\frac{2 (1)}{2 (2 x^{\frac{37}{12}})}$

Cancel the common factor.

$\frac{2 \cdot 1}{2 (2 x^{\frac{37}{12}})}$

Rewrite the expression.

$\frac{1}{2 x^{\frac{37}{12}}}$