Question

Which expression is equivalent to 64y^18-1000z^6?

Answer 1

The expression (4y^6)^3 - (10z^2)^3 is equivalent to 64y^18 - 1000z^6.

Step-by-step explanation:

Factor out common powers:

64y^18 = (2^6)(y^3)^6

1000z^6 = (10^3)(z^2)^3

Rewrite the expression with factored terms:

64y^18 - 1000z^6 = (2^6)(y^3)^6 - (10^3)(z^2)^3

Apply power of a power rule:

(a^m)^n = a^(mn)

(2^6)(y^3)^6 = 2^(66) * y^(36) = 2^36 * y^18

(10^3)(z^2)^3 = 10^(33) * z^(2*3) = 10^9 * z^6

Substitute back the simplified terms:

2^36 * y^18 - 10^9 * z^6 = (4y^6)^3 - (10z^2)^3

Therefore, (4y^6)^3 - (10z^2)^3 is the equivalent expression to 64y^18 - 1000z^6. Both expressions involve the difference of cubes of binomials, with one focusing on powers of 4y^6 and the other emphasizing powers of 10z^2.

Answer 2

For this case we must find an expression equivalent to:
`64y ^ {18}-1000z ^ 6`

Then, we take common factor 8:

`8 (64y ^ {18}-125z ^ 6) =`

By definition of power properties we have to:

`(a ^ n) ^ m = a ^ {n * m}`

So:

`8 ((2y ^ 6) ^ 3- (5z ^ 2) ^ 3)`

Since both terms are perfect cubes, we factor using the cube difference formula:

`(a ^ 3-b ^ 3) = (a-b) (a ^ 2 + 2ab + b ^ 2)`

Where:

`a = 2y ^ 6\\b = 5z ^ 2`

So:
`8 (2y ^ 6-5z ^ 2) ((2y ^ 6) ^ 2 + 2 (2y ^ 6) (5z ^ 2) + (5z ^ 2) ^ 2) =\\8 (2y ^ 6-5z ^ 2) (4y ^ {12} + 10y ^ 6z ^ 2 + 25z ^ 4)`

Answer:

`8 (2y ^ 6-5z ^ 2) (4y ^ {12} + 10y ^ 6z ^ 2 + 25z ^ 4)`