Question

Can anyone help? 80 points!

Answer 1

`4x^2y^3\sqrt[3]{x^2y}`

Explanation:

Given expression:

`\left(8x^4y^5\right)^{(2)/(3)}`

Rewrite 4 as (3 + 1) and 5 as (3 + 2):

`\implies \left(8x^((3+1))y^((3+2))\right)^{(2)/(3)}`

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

`\implies \left(8x^3xy^3y^2}\right)^{(2)/(3)}`

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

`\implies 8^{\left((2)/(3)\right)}x^{\left(3 *(2)/(3)\right)}x^{\left((2)/(3)\right)}y^{\left(3 * (2)/(3)\right)}y^{\left(2 * (2)/(3)\right)}}`

`\implies 4x^2x^{(2)/(3)}y^2y^{(4)/(3)}`

`\implies 4x^2y^2x^{(2)/(3)}y^{(4)/(3)}`

Rewrite 4/3 as 1 + 1/3:

`\implies 4x^2y^2x^{(2)/(3)}y^{\left(1+(1)/(3)\right)}`

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

`\implies 4x^2y^2x^{(2)/(3)}y^1y^{(1)/(3)}`

`\implies 4x^2y^2y^1x^{(2)/(3)}y^{(1)/(3)}`

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

`\implies 4x^2y^((2+1))x^{(2)/(3)}y^{(1)/(3)}`

`\implies 4x^2y^3x^{(2)/(3)}y^{(1)/(3)}`

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

`\implies 4x^2y^3\left(x^2\right)^{(1)/(3)}y^{(1)/(3)`

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

`\implies 4x^2y^3\left(x^2y\right)^{(1)/(3)`

`\textsf{Apply exponent rule} \quad a^{(1)/(n)}=\sqrt[n]{a}:`

`\implies 4x^2y^3\sqrt[3]{x^2y}`

Answer 2

• D)
  `4x^2y^3\sqrt[3]{x^2y}`

Explanation:

Given expression:

• 
  `(8x^4y^5)^{(2)/(3)`

Simplify in steps, using below properties of exponents:

• 
  `a^ba^c= a^(b+c)` Product of Powers Property
• 
  `(a^b)^c=a^(bc)` Power of Power Property

Solution:

• 
  `(8x^4y^5)^{(2)/(3) =`
• 
  `(2^3x^4y^5)^{(2)/(3) =`
• 
  `2^{(3*2)/(3)} x^{(4*2)/(3)} y^{(5*2)/(3)} =`
• 
  `2^2x^{(8)/(3)}y^{(10)/(3) =`
• 
  `4x^{2(2)/(3)}y^{3{(1)/(3) =`
• 
  `4x^2x^{(2)/(3)} y^3y^{(1)/(3)} =`
• 
  `4x^2y^3(x^2y)^{(1)/(3)}=`
• 
  `4x^2y^3\sqrt[3]{x^2y}`

Correct choice is D