Simplify 5r^6t^4((4r^3s^2t^4)/(2r^4st^6))^5

$5 r^{6} t^{4} {(\frac{4 r^{3} s^{2} t^{4}}{2 r^{4} s t^{6}})}^{5}$

Step 1

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 r^{3} s^{2} t^{4}$ .

$5 r^{6} t^{4} {(\frac{2 (2 r^{3} s^{2} t^{4})}{2 r^{4} s t^{6}})}^{5}$

Cancel the common factors.

Factor $2$ out of $2 r^{4} s t^{6}$ .

$5 r^{6} t^{4} {(\frac{2 (2 r^{3} s^{2} t^{4})}{2 (r^{4} s t^{6})})}^{5}$

Cancel the common factor.

$5 r^{6} t^{4} {(\frac{2 (2 r^{3} s^{2} t^{4})}{2 (r^{4} s t^{6})})}^{5}$

Rewrite the expression.

$5 r^{6} t^{4} {(\frac{2 r^{3} s^{2} t^{4}}{r^{4} s t^{6}})}^{5}$

Step 2

Cancel the common factor of $r^{3}$ and $r^{4}$ .

Factor $r^{3}$ out of $2 r^{3} s^{2} t^{4}$ .

$5 r^{6} t^{4} {(\frac{r^{3} (2 s^{2} t^{4})}{r^{4} s t^{6}})}^{5}$

Cancel the common factors.

Factor $r^{3}$ out of $r^{4} s t^{6}$ .

$5 r^{6} t^{4} {(\frac{r^{3} (2 s^{2} t^{4})}{r^{3} (r s t^{6})})}^{5}$

Cancel the common factor.

$5 r^{6} t^{4} {(\frac{r^{3} (2 s^{2} t^{4})}{r^{3} (r s t^{6})})}^{5}$

Rewrite the expression.

$5 r^{6} t^{4} {(\frac{2 s^{2} t^{4}}{r s t^{6}})}^{5}$

Step 3

Cancel the common factor of $s^{2}$ and $s$ .

Factor $s$ out of $2 s^{2} t^{4}$ .

$5 r^{6} t^{4} {(\frac{s (2 s t^{4})}{r s t^{6}})}^{5}$

Cancel the common factors.

Factor $s$ out of $r s t^{6}$ .

$5 r^{6} t^{4} {(\frac{s (2 s t^{4})}{s (r t^{6})})}^{5}$

Cancel the common factor.

$5 r^{6} t^{4} {(\frac{s (2 s t^{4})}{s (r t^{6})})}^{5}$

Rewrite the expression.

$5 r^{6} t^{4} {(\frac{2 s t^{4}}{r t^{6}})}^{5}$

Step 4

Cancel the common factor of $t^{4}$ and $t^{6}$ .

Factor $t^{4}$ out of $2 s t^{4}$ .

$5 r^{6} t^{4} {(\frac{t^{4} (2 s)}{r t^{6}})}^{5}$

Cancel the common factors.

Factor $t^{4}$ out of $r t^{6}$ .

$5 r^{6} t^{4} {(\frac{t^{4} (2 s)}{t^{4} (r t^{2})})}^{5}$

Cancel the common factor.

$5 r^{6} t^{4} {(\frac{t^{4} (2 s)}{t^{4} (r t^{2})})}^{5}$

Rewrite the expression.

$5 r^{6} t^{4} {(\frac{2 s}{r t^{2}})}^{5}$

Step 5

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

Apply the product rule to $\frac{2 s}{r t^{2}}$ .

$5 r^{6} t^{4} \frac{{(2 s)}^{5}}{{(r t^{2})}^{5}}$

Apply the product rule to $2 s$ .

$5 r^{6} t^{4} \frac{2^{5} s^{5}}{{(r t^{2})}^{5}}$

Apply the product rule to $r t^{2}$ .

$5 r^{6} t^{4} \frac{2^{5} s^{5}}{r^{5} {(t^{2})}^{5}}$

Step 6

Simplify the expression.

Raise $2$ to the power of $5$ .

$5 r^{6} t^{4} \frac{32 s^{5}}{r^{5} {(t^{2})}^{5}}$

Multiply the exponents in ${(t^{2})}^{5}$ .

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

$5 r^{6} t^{4} \frac{32 s^{5}}{r^{5} t^{2 \cdot 5}}$

Multiply $2$ by $5$ .

$5 r^{6} t^{4} \frac{32 s^{5}}{r^{5} t^{10}}$

Step 7

Cancel the common factor of $r^{5} t^{4}$ .

Factor $r^{5} t^{4}$ out of $5 r^{6} t^{4}$ .

$r^{5} t^{4} (5 r) \frac{32 s^{5}}{r^{5} t^{10}}$

Factor $r^{5} t^{4}$ out of $r^{5} t^{10}$ .

$r^{5} t^{4} (5 r) \frac{32 s^{5}}{r^{5} t^{4} (t^{6})}$

Cancel the common factor.

$r^{5} t^{4} (5 r) \frac{32 s^{5}}{r^{5} t^{4} t^{6}}$

Rewrite the expression.

$5 r \frac{32 s^{5}}{t^{6}}$

Step 8

Combine $5$ and $\frac{32 s^{5}}{t^{6}}$ .

$r \frac{5 (32 s^{5})}{t^{6}}$

Step 9

Multiply $32$ by $5$ .

$r \frac{160 s^{5}}{t^{6}}$

Step 10

Combine $r$ and $\frac{160 s^{5}}{t^{6}}$ .

$\frac{r (160 s^{5})}{t^{6}}$

Step 11

Move $160$ to the left of $r$ .

$\frac{160 r s^{5}}{t^{6}}$

Simplify 5r^6t^4((4r^3s^2t^4)/(2r^4st^6))^5

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