Question

\frac{2x^3t^3}{xt^2}\:\cdot \frac{\left(3xt\right)^3}{9x^2t}

Answer 1

`6x^(3)t^(3)`

Explanation:

Given expression:

`(2x^3t^3)/(xt^2)\:\cdot (\left(3xt\right)^3)/(9x^2t)`

`\textsf{Apply exponent rule} \quad (ab)^c=a^c \cdot b^c:`

`\implies (2x^3t^3)/(xt^2)\:\cdot (3^3x^3t^3)/(9x^2t)`

`\implies (2x^3t^3)/(xt^2)\:\cdot (27x^3t^3)/(9x^2t)`

`\textsf{Apply the fraction rule} \quad (a)/(b) \cdot (c)/(d)=(ac)/(bd):`

`\implies (2x^3t^327x^3t^3)/(xt^29x^2t)`

Collect like terms:

`\implies (2 \cdot 27x^3x^3t^3t^3)/(9x^2xt^2t)`

Multiply the numbers 2 and 27:

`\implies (54x^3x^3t^3t^3)/(9x^2xt^2t)`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies (54x^(3+3)t^(3+3))/(9x^(2+1)t^(2+1))`

`\implies (54x^6t^6)/(9x^3t^3)`

Separate like terms:

`\implies (54)/(9) \cdot (x^6)/(x^3)\cdot (t^6)/(t^3)`

Divide the numbers 54 and 9:

`\implies 6 \cdot (x^6)/(x^3)\cdot (t^6)/(t^3)`

`\textsf{Apply exponent rule} \quad (a^b)/(a^c)=a^(b-c):`

`\implies 6 \cdot x^(6-3)\cdot t^(6-3)`

`\implies 6 \cdot x^(3)\cdot t^(3)`

`\implies 6x^(3)t^(3)`