Simplify ((x^(1/2)y^-2)/(yx^(-7/4)))^4

${(\frac{x^{\frac{1}{2}} y^{- 2}}{y x^{- \frac{7}{4}}})}^{4}$

Step 1

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

${(\frac{x^{\frac{1}{2}}}{y x^{- \frac{7}{4}} y^{2}})}^{4}$

Step 2

Multiply $y$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

${(\frac{x^{\frac{1}{2}}}{y^{2} y x^{- \frac{7}{4}}})}^{4}$

Multiply $y^{2}$ by $y$ .

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

${(\frac{x^{\frac{1}{2}}}{y^{2} y^{1} x^{- \frac{7}{4}}})}^{4}$

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

${(\frac{x^{\frac{1}{2}}}{y^{2 + 1} x^{- \frac{7}{4}}})}^{4}$

Add $2$ and $1$ .

${(\frac{x^{\frac{1}{2}}}{y^{3} x^{- \frac{7}{4}}})}^{4}$

Step 3

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

${(\frac{x^{\frac{1}{2}} x^{\frac{7}{4}}}{y^{3}})}^{4}$

Step 4

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{7}{4}}$ by adding the exponents.

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

${(\frac{x^{\frac{1}{2} + \frac{7}{4}}}{y^{3}})}^{4}$

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

${(\frac{x^{\frac{1}{2} \cdot \frac{2}{2} + \frac{7}{4}}}{y^{3}})}^{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Add $2$ and $7$ .

${(\frac{x^{\frac{9}{4}}}{y^{3}})}^{4}$

Step 5

Apply the product rule to $\frac{x^{\frac{9}{4}}}{y^{3}}$ .

$\frac{{(x^{\frac{9}{4}})}^{4}}{{(y^{3})}^{4}}$

Step 6

Multiply the exponents in ${(x^{\frac{9}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{\frac{9}{4} \cdot 4}}{{(y^{3})}^{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{x^{\frac{9}{4} \cdot 4}}{{(y^{3})}^{4}}$

Rewrite the expression.

$\frac{x^{9}}{{(y^{3})}^{4}}$

Step 7

Multiply the exponents in ${(y^{3})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{9}}{y^{3 \cdot 4}}$

Multiply $3$ by $4$ .

$\frac{x^{9}}{y^{12}}$

Simplify ((x^(1/2)y^-2)/(yx^(-7/4)))^4

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