Question

1. Simplify (4xy^2)^3(xy)^5

2. Simplify (2t^–3)^3(0.4r)^2
Thank you whoever answers so much!,

Answer 1

The rules of exponents are:
multiplication=add the exponents
x^2.x^3=x^5
power=multiply the exponents
(x^2)^3=x^6

1. (4xy^2)^3(xy)^5
(64x^3y^6)(x^5y^5)
64x^8y^11
2. (2t^-3)^3(0.4r)^2
(8t^-9)(0.16r^2)
0.16r^2/8t^9 or r^2/50t^9

Answer 2

Note main rules:

`1.\quad (a\cdot b)^n=a^n\cdot b^n,\\ \\2.\quad a^m\cdot a^n=a^(m+n),\\ \\3.\quad (a^m)^n=a^(mn).`

1. Consider expression
`(4xy^2)^3(xy)^5.` Using previous rules:

• 1 step:
  `(4xy^2)^3=4^3\cdot x^3\cdot (y^2)^3=64x^3y^(2\cdot 3)=64x^3y^6;`
• 2 step:
  `(xy)^5=x^5\cdot y^5;`
• 3 step:
  `(4xy^2)^3(xy)^5=64x^3y^6x^5y^5=64x^(3+5)y^(6+5)=64x^8y^(11).`

2. Consider expression
`(2t^(-3))^3(0.4r)^2.` Using previous rules:

• 1 step:
  `(2t^(-3))^3=2^3\cdot (t^(-3))^3=8t^(-3\cdot 3)=8t^(-9);`
• 2 step:
  `(0.4r)^2=0.4^2\cdot r^2=0.16r^2;`
• 3 step:
  `(2t^(-3))^3(0.4r)^2=8t^(-9)\cdot 0.16r^2=8\cdot 0.16t^(-9)r^2=1.28t^(-9)r^2.`

Answer:

1. 
   `64x^8y^(11);`
2. 
   `1.28t^(-9)r^2.`