Find the value of (m5)3 n2 + n6 m / nm4 - m7, if n = 4 and m = 1.

Step 1: Recall the difference of cubes formula
$n^3 - m^3 = (n - m)(n^2 + nm + m^2).$
Thus,
$\frac{n^3 - m^3}{n - m} = n^2 + nm + m^2.$

Step 2: Substitute $n = 4$ and $m = 1$:
$n^2 = 4^2 = 16,$
$nm = 4 \cdot 1 = 4,$
$m^2 = 1^2 = 1.$

Step 3: Add the terms:
$16 + 4 + 1 = 21.$

21

Step 1: Compute $r - s$:
$r - s = 3 - 1 = 2.$

Step 2: Compute $t - s$:
$t - s = 2 - 1 = 1.$

Step 3: Compute the numerator $(r - s)^3 - (t - s)^2$:
$(r - s)^3 = 2^3 = 8,$
$(t - s)^2 = 1^2 = 1,$
$8 - 1 = 7.$

Step 4: Divide by the denominator $r - s$:
$\frac{7}{2}.$

\dfrac{7{2}}

Step 1: Factor the expression algebraically:
$(ab)^3 + (bc)^3 = a^3 b^3 + b^3 c^3 = b^3 (a^3 + c^3),$
$\frac{(ab)^3 + (bc)^3}{abc} = \frac{b^3 (a^3 + c^3)}{abc} = \frac{b^2 (a^3 + c^3)}{ac}.$

Step 2: Substitute $a=5$, $b=2$, $c=3$:
$a^3 = 5^3 = 125,$
$c^3 = 3^3 = 27,$
$a^3 + c^3 = 125 + 27 = 152,$
$b^2 = 2^2 = 4,$
$ac = 5 \cdot 3 = 15.$

Step 3: Multiply and divide:
$4 \cdot 152 = 608,$
$\frac{608}{15}.$

\dfrac{608{15}}