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Let's solve each part of the problem using suitable identities:

### (i) \((-3)^3 - 4^3 + 7^3\)
We can use the identity for the sum and difference of cubes: 
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \]
if \(a + b + c = 0\).

Here, \( a = -3 \), \( b = -4 \), \( c = 7 \).

Since \( a + b + c = -3 - 4 + 7 = 0 \), we have:
\[
(-3)^3 + (-4)^3 + 7^3 = 3(-3)(-4)(7) = 252
\]

### (ii) \((10 - 9)^3\)
This simplifies directly to:
\[
(1)^3 = 1
\]

### (iii) \((0.7)^3 + (-0.8)^3 + (0.1)^3\)
Again using the identity \( a + b + c = 0 \).

Here, \( a = 0.7 \), \( b = -0.8 \), \( c = 0.1 \).

Since \( 0.7 - 0.8 + 0.1 = 0 \), we have:
\[
(0.7)^3 + (-0.8)^3 + (0.1)^3 = 3 \times (0.7)(-0.8)(0.1) = -0.168
\]

### (iv) \(3^3 + (-4)^3 + (1)^3 - (-3)^3 + 4^3\)
Let's sum up the cubes directly:

\[
3^3 = 27, \quad (-4)^3 = -64, \quad 1^3 = 1, \quad (-3)^3 = -27, \quad 4^3 = 64
\]

\[
27 - 64 + 1 + 27 + 64 = 55
\]

So, the evaluations are:

1. **(i) 252**
2. **(ii) 1**
3. **(iii) -0.168**
4. **(iv) 55**

Would you like any specific details or explanations for any of these steps? 

Here are 5 related questions you might find helpful:
1. How does the sum of cubes identity work?
2. Can you show the derivation of the cube identity formula?
3. What happens if \( a + b + c \neq 0 \) in the identity?
4. How can you calculate negative cube roots easily?
5. Can you apply these identities to more complex polynomials?

**Tip:** For sums involving cubes, always check if \( a + b + c = 0 \) to simplify calculations using the cube identity!

Evaluate using suitable identity: (i) (-3)^3 - 4^3 + 7^3, (ii) (10 - 9)^3, (iii) (0.7)^3 + (-0.8)^3 + (0.1)^3, (iv) 3^3 + (-4)^3 + (1)^3 - (-3)^3 + 4^3

Learn how to evaluate cubic expressions like (-3)^3 - 4^3 + 7^3 using the sum and difference of cubes identity. Includes step-by-step solutions for other cubic problems, such as (0.7)^3 + (-0.8)^3 + (0.1)^3, making it perfect for high school students.

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