Certainly! Let's simplify each expression step by step:
### Part (a): [tex]\((2x)^3\)[/tex]
To simplify [tex]\((2x)^3\)[/tex]:
1. Recognize that [tex]\((a \cdot b)^3 = a^3 \cdot b^3\)[/tex].
2. Here, [tex]\(a = 2\)[/tex] and [tex]\(b = x\)[/tex].
3. So, [tex]\((2x)^3 = 2^3 \cdot x^3\)[/tex].
4. Calculate [tex]\(2^3\)[/tex]:
[tex]\[
2^3 = 2 \times 2 \times 2 = 8
\][/tex]
5. Therefore:
[tex]\[
(2x)^3 = 8x^3
\][/tex]
### Part (b): [tex]\((-x)^3\)[/tex]
To simplify [tex]\((-x)^3\)[/tex]:
1. Recognize that [tex]\((-a)^3 = -a^3\)[/tex].
2. Here, [tex]\(a = x\)[/tex].
3. So, [tex]\((-x)^3 = -(x^3)\)[/tex].
4. Therefore:
[tex]\[
(-x)^3 = -x^3
\][/tex]
### Part (c): [tex]\((-3m)^3\)[/tex]
To simplify [tex]\((-3m)^3\)[/tex]:
1. Again use the property [tex]\((a \cdot b)^3 = a^3 \cdot b^3\)[/tex].
2. Here, [tex]\(a = -3\)[/tex] and [tex]\(b = m\)[/tex].
3. So, [tex]\((-3m)^3 = (-3)^3 \cdot m^3\)[/tex].
4. Calculate [tex]\((-3)^3\)[/tex]:
[tex]\[
(-3)^3 = -3 \times -3 \times -3 = -27
\][/tex]
5. Therefore:
[tex]\[
(-3m)^3 = -27m^3
\][/tex]
### Part (d): [tex]\((2x^3)^3\)[/tex]
To simplify [tex]\((2x^3)^3\)[/tex]:
1. Use the property [tex]\((a \cdot b)^c = a^c \cdot b^c\)[/tex].
2. Here, [tex]\(a = 2\)[/tex] and [tex]\(b = x^3\)[/tex] with [tex]\(c = 3\)[/tex].
3. So, [tex]\((2x^3)^3 = 2^3 \cdot (x^3)^3\)[/tex].
4. Calculate [tex]\(2^3\)[/tex]:
[tex]\[
2^3 = 8
\][/tex]
5. Simplify [tex]\((x^3)^3\)[/tex]:
[tex]\[
(x^3)^3 = x^{3 \cdot 3} = x^9
\][/tex]
6. Therefore:
[tex]\[
(2x^3)^3 = 8 \cdot x^9 = 8x^9
\][/tex]
In summary, the simplified expressions are:
(a) [tex]\((2x)^3 = 8x^3\)[/tex]
(b) [tex]\((-x)^3 = -x^3\)[/tex]
(c) [tex]\((-3m)^3 = -27m^3\)[/tex]
(d) [tex]\((2x^3)^3 = 8x^9\)[/tex]
