# Test: Algebraic expressions I - Challenging

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Question 1:   Simplify: $\frac{{\left(\sqrt{x}+\sqrt{y}\right)}^{2}+{\left(\sqrt{x}-\sqrt{y}\right)}^{2}}{2\left({x}^{2}-{y}^{2}\right)}÷\frac{1}{{x}^{3}-{y}^{3}}-3xy$
${x}^{2}-{y}^{2}$
${\left(x-y\right)}^{2}$
$x-y$
$1$
Question 2:   Divide ${x}^{4}-7{x}^{3}+5{x}^{2}+6x-7$ by $x+1$
${x}^{3}-8{x}^{2}+10x-7$
${x}^{3}-8{x}^{2}-7$
${x}^{3}-8{x}^{2}+13x-7$
${x}^{3}-8{x}^{2}+13x-5$
Question 3:   Simplify: $\frac{\sqrt{x-2-2\sqrt{x-3}}}{\sqrt{x-3}-1}$
$1$
$\left\{\begin{array}{l}-1,x\in \left[3;4\right)\\ 1,x\in \left(4;+\infty \right)\end{array}$
$\left\{\begin{array}{l}-1,x\in \left[3;4\right)\\ 1,x\in \left[4;+\infty \right)\end{array}$
$0$
Question 4:   Factorise: ${x}^{3}-4{x}^{2}+8x-5$
$\left(x-1\right)\left({x}^{2}-2x+5\right)$
$\left(x-1\right)\left({x}^{2}-3x+2\right)$
$\left(x-1\right)\left(x-2\right)\left(x-3\right)$
$\left(x-1\right)\left({x}^{2}-3x+5\right)$
Question 5:   Simplify: $\sqrt[3]{18{x}^{4}}÷\sqrt[3]{12{x}^{7}}$
$\frac{2}{x}$
$\frac{4}{x}$
$\frac{2}{\sqrt[3]{x}}$
$x$
Question 6:   Simplify: $\frac{{a}^{1}+{a}^{2}+\dots +{a}^{1000}}{{a}^{-1}+{a}^{-2}+\dots +{a}^{-1000}}$
$\frac{1}{{a}^{1001}}$
${a}^{1000}$
${a}^{1001}$
$\frac{1}{{a}^{1000}}$
Question 7:   Factorise: ${x}^{3}-3x+2$
$\left(x-1\right){\left(x+2\right)}^{2}$
$\left(x-1\right)\left({x}^{2}+x-2\right)$
${\left(x-1\right)}^{2}\left(x+2\right)$
$\left(x-1\right)\left(x+2\right)$
Question 8:   Simplify: $\frac{{x}^{7}+{x}^{14}}{{x}^{-7}+{x}^{-14}}$
$\frac{1}{{x}^{21}}$
${x}^{21}$
$\frac{1}{{x}^{7}}$
${x}^{20}$
Question 9:   Simplify: $\frac{1}{{\left(x+3\right)}^{2}}\left(\frac{1}{{x}^{2}}+\frac{1}{9}\right)+\frac{2}{{\left(x+3\right)}^{3}}\left(\frac{1}{x}+\frac{1}{3}\right)$
$9{x}^{2}$
$\frac{1}{3x}$
$\frac{1}{9{x}^{2}}$
$\frac{1}{9x}$
Question 10:   Reduce: $\frac{{\left(x-2\right)}^{2}\left({x}^{2}-9\right)}{\left({x}^{3}-27\right)\left({x}^{2}-4\right)}$
$\frac{x-2}{x+2}$
$\frac{\left(x-2\right)\left(x+3\right)}{\left(x+2\right)\left({x}^{2}+3x+9\right)}$
$\frac{x-3}{{x}^{2}+3x+9}$
$\frac{x-2}{{\left(x-3\right)}^{2}}$