Given that a + b = 9 and a2 + b2 = 71, find the value of (ab)3. (b) Factorise completely. x2 - x − 20 (c) T…

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Given that a + b = 9 and a2 + b2 = 71, find the value of (ab)3. (b) Factorise completely. x2 - x − 20 (c) T = 2pisqrt(l/g). Find T when l = 22.8 and g = 9.8. (d) Express as a single fraction in its simplest form. (a+2)/4 + (a+3)/5

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a) Step 1: Use the algebraic identity (a+b)² = a² + b² + 2ab. Substitute the given values a+b=9 and a²+b²=71 into the identity. (9)² = 71 + 2ab Step 2: Solve for ab. 81 = 71 + 2ab; 81 − 71 = 2ab; 10 = 2ab; ab = 10/2; ab = 5 Step 3: Calculate (ab)³. (ab)³ = (5)³; (ab)³ = 5 × 5 × 5; (ab)³ = 125 The value of (ab)³ is 125. b) Step 1: To factorise x² − x − 20, find two numbers that multiply to −20 and add to −1. The numbers are −5 and 4. Step 2: Write the expression in factored form. x² − x − 20 = (x−5)(x+4) The factorised expression is (x−5)(x+4). c) Step 1: Substitute l = 22.8 and g = 9.8 into the formula T = 2π√(l/g). T = 2π√(22.8/9.8) Step 2: Calculate the value inside the square root. T = 2π√(2.3265306...) Step 3: Calculate the square root and then multiply by 2π. T = 2π × 1.5252969... T = 2 × 3.14159265... × 1.5252969... T = 9.5839... Rounding to three significant figures: T ≈ 9.58 The value of T is 9.58. d) Step 1: Find a common denominator for the fractions (a+2)/(4) and (a+3)/(5). The least common multiple of 4 and 5 is 20. Step 2: Rewrite each fraction with the common denominator. (a+2)/(4) = (5(a+2))/(5 × 4) = (5a+10)/(20); (a+3)/(5) = (4(a+3))/(4 × 5) = (4a+12)/(20) Step 3: Add the two fractions. (5a+10)/(20) + (4a+12)/(20) = ((5a+10) + (4a+12))/(20) Step 4: Combine like terms in the numerator. (5a+4a+10+12)/(20) = (9a+22)/(20) The expression as a single fraction is (9a+22)/(20).

Given that a + b = 9 and a2 + b2 = 71, find the value of (ab)3. (b) Factorise completely. x2 - x - 20 (c) T = 2pisqrt(l/g). Find T when l = 22.8 and g = 9.8. (d) Express as a single fraction in its simplest form. (a+2)/4 + (a+3)/5

Answers

Step-by-step solution

Part a· 3 steps

Use the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$ .

Solve for $ab$ .

Calculate $(ab)^3$ .

Part b· 2 steps(x-5)(x+4)

To factorise $x^2 - x - 20$ , find two numbers that multiply to -20 and add to -1.

Write the expression in factored form.

Part c· 3 steps
$\text{9.58}$

Substitute $l = 22.8$ and $g = 9.8$ into the formula $T = 2\pi\sqrt{\frac{l}{g}}$ .

Calculate the value inside the square root.

Calculate the square root and then multiply by $2\pi$ .

Part d· 4 steps
$\frac{9a+22}{20}$

Find a common denominator for the fractions $\frac{a+2}{4}$ and $\frac{a+3}{5}$ .

Rewrite each fraction with the common denominator.

Add the two fractions.

Combine like terms in the numerator.

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