(b) Simplify the following and write your anirwens as single fractions: \( \begin{array}{lll}\text { (1) }\left(x+\frac{1}{3}\right)\left(x-\frac{1}{3}\right) & \text { (2) }\left(2 x+\frac{1}{3} y\right)\left(3 x-\frac{1}{2} y\right) \text { (3) }\left(x-\frac{1}{4}\right)^{2} \\ \text { (4) }\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right) & \text { (5) }\left(x+\frac{1}{x}\right)\left(x^{2}-1+\frac{1}{x^{2}}\right) \text { (6) }\left(\frac{x}{2}+\frac{2}{x}\right)^{2}\end{array} \)

We wish to simplify each expression and write our answer as a single fraction (or in a “combined” form). Let’s work through each one. ───────────────────────────── (1) Simplify (x + 1/3)(x – 1/3) Recognize this as a difference of two squares:   x² – (1/3)² = x² – 1/9. ───────────────────────────── (2) Simplify (2x + (1/3)y)(3x − (1/2)y) Multiply term‐by‐term:  • 2x · 3x = 6x²  • 2x · (–1/2)y = –x y  • (1/3)y · 3x = x y  • (1/3)y · (–1/2)y = –y²/6 The two middle terms cancel (–xy + xy = 0), leaving   6x² − y²/6. To write as a single fraction, note that 6x² = (36x²)/6 so that   (36x² − y²)/6. ───────────────────────────── (3) Simplify (x − 1/4)² Expand using the square formula:   (x)² − 2·x·(1/4) + (1/4)² = x² − (x/2) + 1/16. ───────────────────────────── (4) Simplify (x + 1/x)(x − 1/x) Again, this is a difference of squares (with a twist):   = x² − (1/x)² = x² − 1/x². To combine into a single fraction, write   = (x⁴ − 1)/x². ───────────────────────────── (5) Simplify (x + 1/x)(x² − 1 + 1/x²) One convenient approach is to first expand:  • Multiply x by each term: x·x² = x³, x·(–1) = –x, x·(1/x²)= 1/x.  • Multiply 1/x by each term: (1/x)·x² = x, (1/x)·(–1) = –1/x, (1/x)·(1/x²)= 1/x³. Then add:   x³ – x + 1/x + x – 1/x + 1/x³ = x³ + 1/x³. To express as a single fraction, write   = (x⁶ + 1)/x³. ───────────────────────────── (6) Simplify (x/2 + 2/x)² First, expand the square:   = (x/2)² + 2·(x/2)(2/x) + (2/x)²   = x²/4 + 2·( (2x)/(2x)) + 4/x²   = x²/4 + 2 + 4/x². To combine these into a single fraction, use the least common denominator 4x²:   = (x²/4)*(x²/x²) + 2*(4x²/4x²) + (4/x²)*(4/4)   = (x⁴)/(4x²) + (8x²)/(4x²) + (16)/(4x²)   = (x⁴ + 8x² + 16)/(4x²). Notice the numerator factors as (x² + 4)², so an acceptable answer is   = (x² + 4)²/(4x²). ───────────────────────────── Final Answers (1) x² – 1/9 (2) (36x² – y²)/6 (3) x² – (x/2) + 1/16 (4) (x⁴ – 1)/x² (5) (x⁶ + 1)/x³ (6) (x² + 4)²/(4x²) Any answer equivalent to the ones above is acceptable.