Simplify (49x^2y)^(1/2)(27x^6y^(3/2))^(1/3)

${(49 x^{2} y)}^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $49 x^{2} y$ .

${(49 x^{2})}^{\frac{1}{2}} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Apply the product rule to $49 x^{2}$ .

$49^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Step 2

Simplify the expression.

Rewrite $49$ as $7^{2}$ .

${(7^{2})}^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

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

$7^{2 (\frac{1}{2})} {(x^{2})}^{\frac{1}{2}} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Step 3

Cancel the common factor of $2$ .

Cancel the common factor.

$7^{2 (\frac{1}{2})} {(x^{2})}^{\frac{1}{2}} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Rewrite the expression.

$7^{1} {(x^{2})}^{\frac{1}{2}} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Step 4

Evaluate the exponent.

$7 {(x^{2})}^{\frac{1}{2}} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Step 5

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

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

$7 x^{2 (\frac{1}{2})} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$7 x^{2 (\frac{1}{2})} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Rewrite the expression.

$7 x^{1} y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Step 6

Simplify.

$7 x y^{\frac{1}{2}} {(27 x^{6} y^{\frac{3}{2}})}^{\frac{1}{3}}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $27 x^{6} y^{\frac{3}{2}}$ .

$7 x y^{\frac{1}{2}} ({(27 x^{6})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

Apply the product rule to $27 x^{6}$ .

$7 x y^{\frac{1}{2}} (27^{\frac{1}{3}} {(x^{6})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

Step 8

Simplify the expression.

Rewrite $27$ as $3^{3}$ .

$7 x y^{\frac{1}{2}} ({(3^{3})}^{\frac{1}{3}} {(x^{6})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

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

$7 x y^{\frac{1}{2}} (3^{3 (\frac{1}{3})} {(x^{6})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

Step 9

Cancel the common factor of $3$ .

Cancel the common factor.

$7 x y^{\frac{1}{2}} (3^{3 (\frac{1}{3})} {(x^{6})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

Rewrite the expression.

$7 x y^{\frac{1}{2}} (3^{1} {(x^{6})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

Step 10

Evaluate the exponent.

$7 x y^{\frac{1}{2}} (3 {(x^{6})}^{\frac{1}{3}} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

Step 11

Multiply the exponents in ${(x^{6})}^{\frac{1}{3}}$ .

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

$7 x y^{\frac{1}{2}} (3 x^{6 (\frac{1}{3})} {(y^{\frac{3}{2}})}^{\frac{1}{3}})$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 12

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Step 13

Multiply the exponents in ${(y^{\frac{3}{2}})}^{\frac{1}{3}}$ .

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 14

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

Move $y^{\frac{1}{2}}$ .

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

$7 x^{3} y^{\frac{2}{2}} \cdot 3$

Divide $2$ by $2$ .

$7 x^{3} y^{1} \cdot 3$

Step 15

Simplify $7 x^{3} y^{1} \cdot 3$ .

$7 x^{3} y \cdot 3$

Step 16

Multiply $3$ by $7$ .

$21 x^{3} y$

Simplify (49x^2y)^(1/2)(27x^6y^(3/2))^(1/3)

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