Simplify ((x^(1/2)*x^(5/12))^4)÷(x^(2/3))

${(x^{\frac{1}{2}} \cdot x^{\frac{5}{12}})}^{4} \div x^{\frac{2}{3}}$

Step 1

Rewrite the division as a fraction.

$\frac{{(x^{\frac{1}{2}} \cdot x^{\frac{5}{12}})}^{4}}{x^{\frac{2}{3}}}$

Step 2

Simplify the numerator.

Apply the product rule to $x^{\frac{1}{2}} \cdot x^{\frac{5}{12}}$ .

$\frac{{(x^{\frac{1}{2}})}^{4} \cdot {(x^{\frac{5}{12}})}^{4}}{x^{\frac{2}{3}}}$

Multiply the exponents in ${(x^{\frac{1}{2}})}^{4}$ .

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

$\frac{x^{\frac{1}{2} \cdot 4} \cdot {(x^{\frac{5}{12}})}^{4}}{x^{\frac{2}{3}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{x^{\frac{1}{2} \cdot (2 (2))} \cdot {(x^{\frac{5}{12}})}^{4}}{x^{\frac{2}{3}}}$

Cancel the common factor.

$\frac{x^{\frac{1}{2} \cdot (2 \cdot 2)} \cdot {(x^{\frac{5}{12}})}^{4}}{x^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{x^{2} \cdot {(x^{\frac{5}{12}})}^{4}}{x^{\frac{2}{3}}}$

Multiply the exponents in ${(x^{\frac{5}{12}})}^{4}$ .

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

$\frac{x^{2} \cdot x^{\frac{5}{12} \cdot 4}}{x^{\frac{2}{3}}}$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

$\frac{x^{2} \cdot x^{\frac{5}{4 (3)} \cdot 4}}{x^{\frac{2}{3}}}$

Cancel the common factor.

$\frac{x^{2} \cdot x^{\frac{5}{4 \cdot 3} \cdot 4}}{x^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{x^{2} \cdot x^{\frac{5}{3}}}{x^{\frac{2}{3}}}$

Step 3

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

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

$\frac{x^{2 + \frac{5}{3}}}{x^{\frac{2}{3}}}$

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

$\frac{x^{2 \cdot \frac{3}{3} + \frac{5}{3}}}{x^{\frac{2}{3}}}$

Combine $2$ and $\frac{3}{3}$ .

$\frac{x^{\frac{2 \cdot 3}{3} + \frac{5}{3}}}{x^{\frac{2}{3}}}$

Combine the numerators over the common denominator.

$\frac{x^{\frac{2 \cdot 3 + 5}{3}}}{x^{\frac{2}{3}}}$

Simplify the numerator.

Multiply $2$ by $3$ .

$\frac{x^{\frac{6 + 5}{3}}}{x^{\frac{2}{3}}}$

Add $6$ and $5$ .

$\frac{x^{\frac{11}{3}}}{x^{\frac{2}{3}}}$

Step 4

Move $x^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{\frac{11}{3}} x^{- \frac{2}{3}}$

Step 5

Multiply $x^{\frac{11}{3}}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

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

$x^{\frac{11}{3} - \frac{2}{3}}$

Combine the numerators over the common denominator.

$x^{\frac{11 - 2}{3}}$

Subtract $2$ from $11$ .

$x^{\frac{9}{3}}$

Divide $9$ by $3$ .

$x^{3}$

Simplify ((x^(1/2)*x^(5/12))^4)÷(x^(2/3))

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