Simplify (27x)^(2/3)(1024x)^(1/5)

${(27 x)}^{\frac{2}{3}} {(1024 x)}^{\frac{1}{5}}$

Step 1

Simplify the expression.

Apply the product rule to $27 x$ .

$27^{\frac{2}{3}} x^{\frac{2}{3}} {(1024 x)}^{\frac{1}{5}}$

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

${(3^{3})}^{\frac{2}{3}} x^{\frac{2}{3}} {(1024 x)}^{\frac{1}{5}}$

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

$3^{3 (\frac{2}{3})} x^{\frac{2}{3}} {(1024 x)}^{\frac{1}{5}}$

Step 2

Cancel the common factor of $3$ .

Cancel the common factor.

$3^{3 (\frac{2}{3})} x^{\frac{2}{3}} {(1024 x)}^{\frac{1}{5}}$

Rewrite the expression.

$3^{2} x^{\frac{2}{3}} {(1024 x)}^{\frac{1}{5}}$

Step 3

Simplify the expression.

Raise $3$ to the power of $2$ .

$9 x^{\frac{2}{3}} {(1024 x)}^{\frac{1}{5}}$

Apply the product rule to $1024 x$ .

$9 x^{\frac{2}{3}} (1024^{\frac{1}{5}} x^{\frac{1}{5}})$

Rewrite using the commutative property of multiplication.

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{2}{3}} x^{\frac{1}{5}}$

Step 4

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

Move $x^{\frac{1}{5}}$ .

$9 \cdot 1024^{\frac{1}{5}} (x^{\frac{1}{5}} x^{\frac{2}{3}})$

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

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{1}{5} + \frac{2}{3}}$

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{1}{5} \cdot \frac{3}{3} + \frac{2}{3}}$

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{1}{5} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{5}{5}}$

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{3}{5 \cdot 3} + \frac{2}{3} \cdot \frac{5}{5}}$

Multiply $5$ by $3$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{3}{15} + \frac{2}{3} \cdot \frac{5}{5}}$

Multiply $\frac{2}{3}$ by $\frac{5}{5}$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{3}{15} + \frac{2 \cdot 5}{3 \cdot 5}}$

Multiply $3$ by $5$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{3}{15} + \frac{2 \cdot 5}{15}}$

Combine the numerators over the common denominator.

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{3 + 2 \cdot 5}{15}}$

Simplify the numerator.

Multiply $2$ by $5$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{3 + 10}{15}}$

Add $3$ and $10$ .

$9 \cdot 1024^{\frac{1}{5}} x^{\frac{13}{15}}$

Step 5

Simplify the expression.

Rewrite $1024$ as $4^{5}$ .

$9 \cdot {(4^{5})}^{\frac{1}{5}} x^{\frac{13}{15}}$

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

$9 \cdot 4^{5 (\frac{1}{5})} x^{\frac{13}{15}}$

Step 6

Cancel the common factor of $5$ .

Cancel the common factor.

$9 \cdot 4^{5 (\frac{1}{5})} x^{\frac{13}{15}}$

Rewrite the expression.

$9 \cdot 4^{1} x^{\frac{13}{15}}$

Step 7

Simplify the expression.

Evaluate the exponent.

$9 \cdot 4 x^{\frac{13}{15}}$

Multiply $9$ by $4$ .

$36 x^{\frac{13}{15}}$

Simplify (27x)^(2/3)(1024x)^(1/5)

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