1. Express in p/q form (where p and q are integers and q≠0): (a) 8 (b) 6-4/√2. 2. Rationalise the denominator of: (a) 7/(2√3+√5) (b) 3√5-√7/(3√3+√2). 3. Find a and b. (1) If (5-3√2... 1. Express in p/q form (where p and q are integers and q≠0): (a) 8 (b) 6-4/√2. 2. Rationalise the denominator of: (a) 7/(2√3+√5) (b) 3√5-√7/(3√3+√2). 3. Find a and b. (1) If (5-3√2)/(5+3√2) = a + b√2 (2) If (6+4√2)/(6+4√2) = a - b√2. 4. Simplify: √(81)-8√343+15√32+√225. 5. If x=2+√3, find the value of x² + 1/x². 6. Write in ascending order: √(3)/√(3), √(3), √(5), √(5). 7. Which is smaller √5 or √2? 8. Simplify: (16/81)^(-3/2) × (64/125)^(3/4). 9. Simplify: 2√8 + 2√32 - 2√2. 10. If x-2 is a factor of p(x)=2x²-2x-3x+7a, find the value of a. 11. For what value of m, x³-2m²x+2=16 is divisible by 4? Read more

Steps to Solve

1. Expressing in ( p/q ) form To express a number in the form ( p/q ), where ( p ) and ( q ) are integers, identify the number and convert it. For example, ( 0.75 ) can be expressed as ( \frac{3}{4} ).

2. Rationalizing the denominator (1a) Given the fraction ( \frac{7}{2+\sqrt{3}} ), multiply the numerator and the denominator by the conjugate of the denominator, ( 2-\sqrt{3} ): $$ \frac{7(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} $$ Calculate the denominator: $$ (2+\sqrt{3})(2-\sqrt{3}) = 4 - 3 = 1 $$ Thus, the result is simply: $$ 7(2-\sqrt{3}) $$

3. Rationalizing the denominator (1b) For ( \frac{3\sqrt{5}-\sqrt{7}}{3+\sqrt{5}} ), multiply by ( 3-\sqrt{5} ) similar to step 2: $$ \frac{(3\sqrt{5}-\sqrt{7})(3-\sqrt{5})}{(3+\sqrt{5})(3-\sqrt{5})} $$ Calculate the denominator: $$ 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 $$ Now, simplify the multiplication in the numerator.

4. Find ( a ) and ( b ) For the expression ( \frac{5-\sqrt{3}}{5+\sqrt{2}} ), multiply by the conjugate: $$ \frac{(5-\sqrt{3})(5-\sqrt{2})}{(5+\sqrt{2})(5-\sqrt{2})} $$ Calculate the denominator: $$ 25 - 2 = 23 $$ The numerator becomes ( 5(5) - 5\sqrt{2} - \sqrt{3}(5) + \sqrt{6} ).

5. Simplification of ( \sqrt{81} - 8\sqrt{343} + 15\sqrt{32} - \sqrt{225} ) Simplify each square root separately:

• ( \sqrt{81} = 9 )
• ( \sqrt{343} = 7 )
• ( \sqrt{32} = 4\sqrt{2} )
• ( \sqrt{225} = 15 )

Combine terms to get the final result.

6. Finding ( x^2 + \frac{1}{x^2} ) Using the identity ( (x + \frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2} ), substitute ( x = 2 + \sqrt{3} ). Then find ( x + \frac{1}{x} ) and square it, subtracting 2 to find the value.

7. Write in ascending order For ( \sqrt[3]{3}, \sqrt[3]{2}, \sqrt{3}, \sqrt{5} ): Calculate approximate decimal values to order them.

8. Comparison of ( (16/81)^{-3/4} \times (64/125)^{-3/4} ) Apply the exponent to each and simplify: $$ (16^{-3/4} \times 81^{3/4}) \times (64^{-3/4} \times 125^{3/4}) $$ Combine the bases under one exponent.

9. Simplifying ( 2\sqrt{8} + 2\sqrt{32} - 2\sqrt{2} ) First, simplify each square root:

• ( 2\sqrt{8} = 4\sqrt{2} )
• ( 2\sqrt{32} = 8\sqrt{2} ) Combine ( 4\sqrt{2} + 8\sqrt{2} - 2\sqrt{2} = 10\sqrt{2} ).

10. Finding value of ( x ) Translate ( x - 2 ) as a factor and equate to previously defined polynomial, factor it out to find roots.