Simplify 6(x^17)^(1/6)*(8x)^(2/3)

$6 {(x^{17})}^{\frac{1}{6}} \cdot {(8 x)}^{\frac{2}{3}}$

Step 1

Multiply the exponents in ${(x^{17})}^{\frac{1}{6}}$ .

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

$6 x^{17 (\frac{1}{6})} \cdot {(8 x)}^{\frac{2}{3}}$

Combine $17$ and $\frac{1}{6}$ .

$6 x^{\frac{17}{6}} \cdot {(8 x)}^{\frac{2}{3}}$

Step 2

Apply the product rule to $8 x$ .

$6 x^{\frac{17}{6}} \cdot (8^{\frac{2}{3}} x^{\frac{2}{3}})$

Step 3

Rewrite using the commutative property of multiplication.

$6 \cdot 8^{\frac{2}{3}} x^{\frac{17}{6}} x^{\frac{2}{3}}$

Step 4

Multiply $x^{\frac{17}{6}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

Move $x^{\frac{2}{3}}$ .

$6 \cdot 8^{\frac{2}{3}} (x^{\frac{2}{3}} x^{\frac{17}{6}})$

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

$6 \cdot 8^{\frac{2}{3}} x^{\frac{2}{3} + \frac{17}{6}}$

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

$6 \cdot 8^{\frac{2}{3}} x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{17}{6}}$

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

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

$6 \cdot 8^{\frac{2}{3}} x^{\frac{2 \cdot 2}{3 \cdot 2} + \frac{17}{6}}$

Multiply $3$ by $2$ .

$6 \cdot 8^{\frac{2}{3}} x^{\frac{2 \cdot 2}{6} + \frac{17}{6}}$

Combine the numerators over the common denominator.

$6 \cdot 8^{\frac{2}{3}} x^{\frac{2 \cdot 2 + 17}{6}}$

Simplify the numerator.

Multiply $2$ by $2$ .

$6 \cdot 8^{\frac{2}{3}} x^{\frac{4 + 17}{6}}$

Add $4$ and $17$ .

$6 \cdot 8^{\frac{2}{3}} x^{\frac{21}{6}}$

Cancel the common factor of $21$ and $6$ .

Factor $3$ out of $21$ .

$6 \cdot 8^{\frac{2}{3}} x^{\frac{3 (7)}{6}}$

Cancel the common factors.

Factor $3$ out of $6$ .

$6 \cdot 8^{\frac{2}{3}} x^{\frac{3 \cdot 7}{3 \cdot 2}}$

Cancel the common factor.

$6 \cdot 8^{\frac{2}{3}} x^{\frac{3 \cdot 7}{3 \cdot 2}}$

Rewrite the expression.

$6 \cdot 8^{\frac{2}{3}} x^{\frac{7}{2}}$

Step 5

Simplify the expression.

Rewrite $8$ as $2^{3}$ .

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

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

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

Step 6

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$6 \cdot 2^{2} x^{\frac{7}{2}}$

Step 7

Simplify the expression.

Raise $2$ to the power of $2$ .

$6 \cdot 4 x^{\frac{7}{2}}$

Multiply $6$ by $4$ .

$24 x^{\frac{7}{2}}$

Simplify 6(x^17)^(1/6)*(8x)^(2/3)

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