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\(\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}\) Simplifying from the innermost radical and progressing outwards we have the given expression \(\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}\) = \(\sqrt{160r^2 + \sqrt{71r^4 + 10r^4}}\) = \(\sqrt{160r^2 + \sqrt{81r^4}}\) \(\sqrt{160r^2 + 9r^2}\) = \(\sqrt{169r^2}\) = 13r

3 log\(_6\)9 + log\(_6\)12 + log\(_6\)64 - log\(_6\)72 = log\(_6\)9\(^3\) + log\(_6\)12 + log\(_6\)64 - log\(_6\)72 log\(_6\)729 + log\(_6\)12 + log\(_6\)64 - log\(_6\)72 log\(_6\)[\(\frac{(729 \times 12 \times 64)}{72}\)] = log\(_6\)7776 = log\(_6\)6\(^5\) = 5log\(_6\)6 = 5 N.B: log\(_6\)6 = 1

Simplify (\(\frac{1}{x^{-1}} \div \frac{1}{y^{-1}}\))-1 = (\(\frac{1}{x^{-1}} \div \frac{1}{y^{-1}}\))-1 = \((x \div y)^{-1}\) = \(\frac{(x^{-1})}{y^{-1}}\) = \(\frac{y}{x}\)

(ab2 - bc2)(a2c - abc) [2(2)2 - (- 2x\(\frac{1}{2}\))] [22(-\(\frac{1}{2}\)) - 2(-2)(-\(\frac{1}{2}\))] [8 = \(\frac{1}{2}\)][-2 - 2] = \(\frac{17}{2}\) x 42 = -34

f(x - 4) = x2 + 2x + 3 To find f(2) = f(x - 4) = f(2) x - 4 = 2 x = 6 f(2) = 62 + 2(6) + 3 = 36 + 12 + 3 = 51

9(x + y)2 - 4(x - y)2 Using difference of two squares which says a2 - b2 = (a + b)(a - b) = 9(x + y)2 - 4(x - y)2 = [3(x + y)]2 - [2(x - y)]-2 = [3(x + y) + 2(x - y) + 2(x - y)][3(x + y) - 2(x - y)] = [3x +3y + 2x - 2y][3x + 3y - 2x + 2y] = (5x + y)(x + 5y)

a2 + b2 = 16 and 2ab = 7 To find all possible values = (a - b)2 + b2 - 2ab Substituting the given values = (a - b)2 = 16 - 7 = 9 (a - b) = \(\pm\)9 = \(\pm\)3 OR a - b = 3, -3

Step 1: Just multiply each of the options by ( x - 1 ) Step 2: Then collect like terms to derive the same equation found in the question. Hope this helps!

x + \(\frac{1}{x}\) = 4, find x2 + \(\frac{1}{x^2}\) = (x + \(\frac{1}{x}\))2 = x2 + \(\frac{1}{x^2}\) + 2 x2 + \(\frac{1}{x^2}\) = ( x + \(\frac{1}{x^2}\))2 - 2 = (4)2 - 2 = 16 - 2 = 14

4x\(^2\) - 4 = (2x)\(^2\) - 2\(^2\) (2x)\(^2\) - 2\(^2\) = a\(^2\) - b\(^2\) = (a - b)\(^2\) - 2ab - 2b\(^2\) (a - b)\(^2\) = a - b\(^2\) + (2ab + 2b)\(^2\) a = 2x and b = 2 2ab + 2b\(^2\) = 2(2x)(2) + 2(2)\(^2\) = 8x + 8 = 8(x + 1)

x - 8\(\sqrt{x}\) + 15 = 0 x + 15 = 8\(\sqrt{x}\) square both sides = (x + 15)2 = (8 \(\sqrt{x}\)2 x2 + 225 + 30x = 64x x2 + 225 + 30x - 64x = 0 x2 - 34x + 225 = 0 (x - 9)(x - 25) = 0 x = 9 or 25

Perimeter. of a rectangle = 2(L + B) = 24 Area of a rectangle = L x B = 35 From above, L + B = 12 L = 12 - B L x B = 35(given) 12 - B)B = 35 12B - B\(^2\) = 35 B\(^2\) - 12B + 35 = 0 B\(^2\) - 7B - 5B + 35 = 0 B(B - 7) - 5(B - 7) = 0 (B-5)(B -7) = 0 B = 5m OR 7m

Let his assistant work for x days ∴ his master worked (x + 10) day. Amount received by master = 40(x + 10), amount got by his assistance = 10x Total amount collected = N2000.00 ∴ 40(x + 10) + 10x = 2000 = 40x + 400 + 10x = 2000 50x + 400 = 2000 50x = 2000 - 400 50x = 1600 x = \(\frac{1600}{50}\) x = 32 days

\(\frac{x}{x + y}\) + \(\frac{y}{x - y}\) - \(\frac{x^2}{x^2 - y^2}\) \(\frac{x}{x + y}\) + \(\frac{y}{x - y}\) - \(\frac{x^2}{(x + y)(x - y}\) = \(\frac{x(x - y) + y(x + y) - x^2}{(x + y)(x - y}\) = \(\frac{x^2 + xy + xy + y^2 - x^2}{(x + y)(x - y}\) = \(\frac{y^2}{(x + y)(x - y)}\) = \(\frac{y^2}{(x^2 - y^2)}\)

Given that x2 + y2 + z2 = 194, calculate z if x = 7 and \(\sqrt{y}\) = 3 x = 7 ∴ x2 = 49 \(\sqrt{y}\) = 3 ∴ y2 = 81 = x2 + y2 + z2 = 194 49 + 81 + z2 = 194 130 + z2 = 194 z2 = 194 - 130 = 64 z = \(\sqrt{64}\) = 8