Answer :
Sure, let's go through each question step-by-step.
### Question 1: Evaluate
#### i) [tex]\( 23^3 - 17^3 \)[/tex]
Let's calculate [tex]\( 23^3 \)[/tex] and [tex]\( 17^3 \)[/tex] first:
[tex]\[ 23^3 = 23 \times 23 \times 23 = 529 \times 23 = 12167 \][/tex]
[tex]\[ 17^3 = 17 \times 17 \times 17 = 289 \times 17 = 4913 \][/tex]
Therefore,
[tex]\[ 23^3 - 17^3 = 12167 - 4913 = 7254 \][/tex]
#### ii) [tex]\( 29^3 - 11^3 \)[/tex]
Let's calculate [tex]\( 29^3 \)[/tex] and [tex]\( 11^3 \)[/tex] first:
[tex]\[ 29^3 = 29 \times 29 \times 29 = 841 \times 29 = 24389 \][/tex]
[tex]\[ 11^3 = 11 \times 11 \times 11 = 121 \times 11 = 1331 \][/tex]
Therefore,
[tex]\[ 29^3 - 11^3 = 24389 - 1331 = 23058 \][/tex]
### Question 2: Simplify
We'll use algebraic identities to simplify the given expressions.
#### i) [tex]\( (4x + 2y)^3 + (4x - 2y)^3 \)[/tex]
Using the identity [tex]\( (a + b)^3 + (a - b)^3 = 2(a^3 + b^3) \)[/tex]:
Let [tex]\( a = 4x \)[/tex] and [tex]\( b = 2y \)[/tex], then
[tex]\[ (4x + 2y)^3 + (4x - 2y)^3 = 2((4x)^3 + (2y)^3) = 2(64x^3 + 8y^3) = 128x^3 + 16y^3 \][/tex]
#### ii) [tex]\( (4x + 2y)^3 - (4x - 2y)^3 \)[/tex]
Using the identity [tex]\( (a + b)^3 - (a - b)^3 = 2a(a^2 + 3b^2) \)[/tex]:
Let [tex]\( a = 4x \)[/tex] and [tex]\( b = 2y \)[/tex], then
[tex]\[ (4x + 2y)^3 - (4x - 2y)^3 = 2(4x)((4x)^2 + 3(2y)^2) \][/tex]
[tex]\[ = 2(4x)(16x^2 + 12y^2) = 8x(16x^2 + 12y^2) = 128x^3 + 96xy^2 \][/tex]
### Question 3: Find the value of [tex]\( a^3 + b^3 \)[/tex]
Given:
[tex]\[ a + b = 10 \][/tex]
[tex]\[ a^2 + b^2 = 58 \][/tex]
We use the identity:
[tex]\[ (a + b)^2 = a^2 + b^2 + 2ab \][/tex]
Let's find [tex]\( ab \)[/tex]:
[tex]\[ (10)^2 = 58 + 2ab \][/tex]
[tex]\[ 100 = 58 + 2ab \][/tex]
[tex]\[ 2ab = 42 \][/tex]
[tex]\[ ab = 21 \][/tex]
Using the identity:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
We already know [tex]\( a + b = 10 \)[/tex], [tex]\( a^2 + b^2 = 58 \)[/tex], and [tex]\( ab = 21 \)[/tex]. Now, calculate:
[tex]\[ a^2 - ab + b^2 = a^2 + b^2 - ab = 58 - 21 = 37 \][/tex]
Therefore,
[tex]\[ a^3 + b^3 = 10 \times 37 = 370 \][/tex]
### Summary of Answers:
1. (i) [tex]\( 23^3 - 17^3 = 7254 \)[/tex]
(ii) [tex]\( 29^3 - 11^3 = 23058 \)[/tex]
2. (i) [tex]\( (4x + 2y)^3 + (4x - 2y)^3 = 128x^3 + 16y^3 \)[/tex]
(ii) [tex]\( (4x + 2y)^3 - (4x - 2y)^3 = 128x^3 + 96xy^2 \)[/tex]
3. [tex]\( a^3 + b^3 = 370 \)[/tex]
### Question 1: Evaluate
#### i) [tex]\( 23^3 - 17^3 \)[/tex]
Let's calculate [tex]\( 23^3 \)[/tex] and [tex]\( 17^3 \)[/tex] first:
[tex]\[ 23^3 = 23 \times 23 \times 23 = 529 \times 23 = 12167 \][/tex]
[tex]\[ 17^3 = 17 \times 17 \times 17 = 289 \times 17 = 4913 \][/tex]
Therefore,
[tex]\[ 23^3 - 17^3 = 12167 - 4913 = 7254 \][/tex]
#### ii) [tex]\( 29^3 - 11^3 \)[/tex]
Let's calculate [tex]\( 29^3 \)[/tex] and [tex]\( 11^3 \)[/tex] first:
[tex]\[ 29^3 = 29 \times 29 \times 29 = 841 \times 29 = 24389 \][/tex]
[tex]\[ 11^3 = 11 \times 11 \times 11 = 121 \times 11 = 1331 \][/tex]
Therefore,
[tex]\[ 29^3 - 11^3 = 24389 - 1331 = 23058 \][/tex]
### Question 2: Simplify
We'll use algebraic identities to simplify the given expressions.
#### i) [tex]\( (4x + 2y)^3 + (4x - 2y)^3 \)[/tex]
Using the identity [tex]\( (a + b)^3 + (a - b)^3 = 2(a^3 + b^3) \)[/tex]:
Let [tex]\( a = 4x \)[/tex] and [tex]\( b = 2y \)[/tex], then
[tex]\[ (4x + 2y)^3 + (4x - 2y)^3 = 2((4x)^3 + (2y)^3) = 2(64x^3 + 8y^3) = 128x^3 + 16y^3 \][/tex]
#### ii) [tex]\( (4x + 2y)^3 - (4x - 2y)^3 \)[/tex]
Using the identity [tex]\( (a + b)^3 - (a - b)^3 = 2a(a^2 + 3b^2) \)[/tex]:
Let [tex]\( a = 4x \)[/tex] and [tex]\( b = 2y \)[/tex], then
[tex]\[ (4x + 2y)^3 - (4x - 2y)^3 = 2(4x)((4x)^2 + 3(2y)^2) \][/tex]
[tex]\[ = 2(4x)(16x^2 + 12y^2) = 8x(16x^2 + 12y^2) = 128x^3 + 96xy^2 \][/tex]
### Question 3: Find the value of [tex]\( a^3 + b^3 \)[/tex]
Given:
[tex]\[ a + b = 10 \][/tex]
[tex]\[ a^2 + b^2 = 58 \][/tex]
We use the identity:
[tex]\[ (a + b)^2 = a^2 + b^2 + 2ab \][/tex]
Let's find [tex]\( ab \)[/tex]:
[tex]\[ (10)^2 = 58 + 2ab \][/tex]
[tex]\[ 100 = 58 + 2ab \][/tex]
[tex]\[ 2ab = 42 \][/tex]
[tex]\[ ab = 21 \][/tex]
Using the identity:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
We already know [tex]\( a + b = 10 \)[/tex], [tex]\( a^2 + b^2 = 58 \)[/tex], and [tex]\( ab = 21 \)[/tex]. Now, calculate:
[tex]\[ a^2 - ab + b^2 = a^2 + b^2 - ab = 58 - 21 = 37 \][/tex]
Therefore,
[tex]\[ a^3 + b^3 = 10 \times 37 = 370 \][/tex]
### Summary of Answers:
1. (i) [tex]\( 23^3 - 17^3 = 7254 \)[/tex]
(ii) [tex]\( 29^3 - 11^3 = 23058 \)[/tex]
2. (i) [tex]\( (4x + 2y)^3 + (4x - 2y)^3 = 128x^3 + 16y^3 \)[/tex]
(ii) [tex]\( (4x + 2y)^3 - (4x - 2y)^3 = 128x^3 + 96xy^2 \)[/tex]
3. [tex]\( a^3 + b^3 = 370 \)[/tex]
