Which expression is equivalent to [tex]$64 y^{18}-1000 z^6$[/tex]?

A. [tex]\left(4 y^6\right)^3-\left(10 z^2\right)^3[/tex]
B. [tex]\left(16 y^6\right)^3-\left(10 z^2\right)^3[/tex]
C. [tex]\left(16 y^6\right)^3-\left(100 z^2\right)^3[/tex]
D. [tex]\left(4 y^6\right)^3-\left(100 z^2\right)^3[/tex]



Answer :

To determine the expression equivalent to [tex]\(64 y^{18}-1000 z^6\)[/tex], we must rewrite the given expression in the form of [tex]\(a^3 - b^3\)[/tex]. Let's break this down step-by-step:

1. Identify possible values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:

Start by recognizing that the given expression is in the form [tex]\(64y^{18} - 1000z^6\)[/tex].

We want to express both terms as cubes:
[tex]\[ a^3 = 64y^{18} \quad \text{and} \quad b^3 = 1000z^6 \][/tex]

2. Express [tex]\(64y^{18}\)[/tex] as a cube:

Notice that:
[tex]\[ 64 = 4^3 \quad \text{and} \quad y^{18} = (y^6)^3 \][/tex]

Hence, we can combine them as:
[tex]\[ 64y^{18} = (4y^6)^3 \][/tex]

3. Express [tex]\(1000z^6\)[/tex] as a cube:

Notice that:
[tex]\[ 1000 = 10^3 \quad \text{and} \quad z^6 = (z^2)^3 \][/tex]

Hence, we can combine them as:
[tex]\[ 1000z^6 = (10z^2)^3 \][/tex]

4. Combining both expressions:

Now we have:
[tex]\[ 64y^{18} = (4y^6)^3 \quad \text{and} \quad 1000z^6 = (10z^2)^3 \][/tex]

Therefore:
[tex]\[ 64y^{18} - 1000z^6 = (4y^6)^3 - (10z^2)^3 \][/tex]

Thus, the expression equivalent to [tex]\(64 y^{18} - 1000 z^6\)[/tex] is:
[tex]\[ \left(4 y^6\right)^3 - \left(10 z^2\right)^3 \][/tex]

Hence, the correct choice is:
[tex]\[ \boxed{\left(4 y^6\right)^3 - \left(10 z^2\right)^3} \][/tex]