Answer :
To solve the given question, let's focus on the expression [tex]\(\frac{x}{y} + \frac{3}{2}\)[/tex] and finding the value of [tex]\(\frac{x^3 + y^3}{x^2 - y^2}\)[/tex].
### Step-by-Step Solution:
1. Given the Expression:
[tex]\[ \frac{x}{y} + \frac{3}{2} \][/tex]
This is an expression in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Without additional specific values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], the expression remains as it is.
2. Expression to Simplify:
[tex]\[ \frac{x^3 + y^3}{x^2 - y^2} \][/tex]
3. Using Algebraic Identities:
We can use the algebraic identities to rewrite and simplify the expression:
- The identity for the sum of cubes:
[tex]\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \][/tex]
- The difference of squares:
[tex]\[ x^2 - y^2 = (x-y)(x+y) \][/tex]
4. Form the Fraction and Simplify:
First, substitute the algebraic identities into the fraction:
[tex]\[ \frac{(x+y)(x^2 - xy + y^2)}{(x-y)(x+y)} \][/tex]
5. Cancel Common Factors:
The [tex]\((x+y)\)[/tex] terms in the numerator and the denominator cancel each other out:
[tex]\[ \frac{x^2 - xy + y^2}{x-y} \][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{x^3 + y^3}{x^2 - y^2}\)[/tex] is [tex]\(\frac{x^2 - xy + y^2}{x-y}\)[/tex].
Given the result from the analysis above, our answer is:
[tex]\[ [x/y + 1.5, (x^3 + y^3)/(x^2 - y^2), (x^3 + y^3)/(x^2 - y^2)] \][/tex]
It matches:
- [tex]\(\frac{x}{y} + \frac{3}{2}\)[/tex]
- [tex]\(\frac{x^3 + y^3}{x^2 - y^2}\)[/tex]
### Answer Recognition:
Based on the given options in part 60:
- The simplified form of the expression does not match any numerical value provided in the options (A, B, C, D).
So, the correct choice for part 60 is:
[tex]\( \boxed{D\text{)}\ None\ of\ these} \)[/tex]
### Part 61:
Given the expression to evaluate:
[tex]\[ 2 + 1 + \left\{ 2 + 1 + \left( 2 + \frac{1}{3} \right) \right\} \][/tex]
1. Evaluate the innermost expression:
[tex]\[ 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \][/tex]
2. Move outward and add:
[tex]\[ 2 + 1 + \left(\frac{7}{3}\right) = 3 + \frac{7}{3} = \frac{9}{3} + \frac{7}{3} = \frac{16}{3} \][/tex]
3. Finally, add the outermost terms:
[tex]\[ 2 + 1 + \left(\frac{16}{3}\right) = 3 + \frac{16}{3} = \frac{9}{3} + \frac{16}{3} = \frac{25}{3} \][/tex]
Therefore, the correct answer for part 61 is:
[tex]\( \boxed{25} \)[/tex]
### Step-by-Step Solution:
1. Given the Expression:
[tex]\[ \frac{x}{y} + \frac{3}{2} \][/tex]
This is an expression in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Without additional specific values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], the expression remains as it is.
2. Expression to Simplify:
[tex]\[ \frac{x^3 + y^3}{x^2 - y^2} \][/tex]
3. Using Algebraic Identities:
We can use the algebraic identities to rewrite and simplify the expression:
- The identity for the sum of cubes:
[tex]\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \][/tex]
- The difference of squares:
[tex]\[ x^2 - y^2 = (x-y)(x+y) \][/tex]
4. Form the Fraction and Simplify:
First, substitute the algebraic identities into the fraction:
[tex]\[ \frac{(x+y)(x^2 - xy + y^2)}{(x-y)(x+y)} \][/tex]
5. Cancel Common Factors:
The [tex]\((x+y)\)[/tex] terms in the numerator and the denominator cancel each other out:
[tex]\[ \frac{x^2 - xy + y^2}{x-y} \][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{x^3 + y^3}{x^2 - y^2}\)[/tex] is [tex]\(\frac{x^2 - xy + y^2}{x-y}\)[/tex].
Given the result from the analysis above, our answer is:
[tex]\[ [x/y + 1.5, (x^3 + y^3)/(x^2 - y^2), (x^3 + y^3)/(x^2 - y^2)] \][/tex]
It matches:
- [tex]\(\frac{x}{y} + \frac{3}{2}\)[/tex]
- [tex]\(\frac{x^3 + y^3}{x^2 - y^2}\)[/tex]
### Answer Recognition:
Based on the given options in part 60:
- The simplified form of the expression does not match any numerical value provided in the options (A, B, C, D).
So, the correct choice for part 60 is:
[tex]\( \boxed{D\text{)}\ None\ of\ these} \)[/tex]
### Part 61:
Given the expression to evaluate:
[tex]\[ 2 + 1 + \left\{ 2 + 1 + \left( 2 + \frac{1}{3} \right) \right\} \][/tex]
1. Evaluate the innermost expression:
[tex]\[ 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \][/tex]
2. Move outward and add:
[tex]\[ 2 + 1 + \left(\frac{7}{3}\right) = 3 + \frac{7}{3} = \frac{9}{3} + \frac{7}{3} = \frac{16}{3} \][/tex]
3. Finally, add the outermost terms:
[tex]\[ 2 + 1 + \left(\frac{16}{3}\right) = 3 + \frac{16}{3} = \frac{9}{3} + \frac{16}{3} = \frac{25}{3} \][/tex]
Therefore, the correct answer for part 61 is:
[tex]\( \boxed{25} \)[/tex]
