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Exercises: Computing with Square Roots

Simplify as far as possible.

$(1 - \sqrt{3}) \cdot (1 + \sqrt{3})$

For this task you need the following basic knowledge: Powers
$(1 - \sqrt{3}) \cdot (1 + \sqrt{3})$ $=$
↓
Use the binomial formulas.
$=$ $(1^{2} - {\sqrt{3}}^{2})$
↓
Square and root cancel.
$=$ $(1 - 3)$
$=$ $- 2$
${(\sqrt{2} - 3 \sqrt{2})}^{2}$

For this task you need the following basic knowledge: Powers
${(\sqrt{2} - 3 \sqrt{2})}^{2}$ $=$
↓
Subtract inside the bracket
$=$ ${(- 2 \sqrt{2})}^{2}$
$=$ ${(- 2)}^{2} \cdot {(\sqrt{2})}^{2}$
↓
Square both parts.
$=$ $4 \cdot 2$
$=$ $8$
$\sqrt{3} \cdot (\frac{1}{6} \sqrt{12} - 3 \sqrt{\frac{1}{27}})$

For this task you need the following basic knowledge: Powers
$\sqrt{3} \cdot (\frac{1}{6} \sqrt{12} - 3 \sqrt{\frac{1}{27}})$ $=$
↓
Pull $\frac{1}{6}$ and $3$ into the root.
$=$ $\sqrt{3} \cdot (\sqrt{\frac{1}{36} \cdot 12} - \sqrt{9 \cdot \frac{1}{27}})$
↓
Multiply inside the roots.
$=$ $\sqrt{3} \cdot (\sqrt{\frac{1}{3}} - \sqrt{\frac{1}{3}})$
↓
Subtract inside the bracket.
$=$ $\sqrt{3} \cdot 0$
$=$ $0$

$(2 \sqrt{108} - 7 \sqrt{54}) : \sqrt{27}$

${(\sqrt{2} - \sqrt{18})}^{2}$

For this task you need the following basic knowledge: Powers
${(\sqrt{2} - \sqrt{18})}^{2}$ $=$
↓
Apply the binomial formulas.
$=$ ${\sqrt{2}}^{2} - 2 \cdot \sqrt{2} \cdot \sqrt{18} + {\sqrt{18}}^{2}$
$=$ $2 - 12 + 18$
$=$ $8$
Alternative Solution
${(\sqrt{2} - \sqrt{18})}^{2}$ $=$ ${(\sqrt{2} - 3 \sqrt{2})}^{2}$
↓
conclude
$=$ ${(- 2 \sqrt{2})}^{2}$
$=$ $(- 2)^{2} \cdot (\sqrt{2})^{2}$
$=$ $4 \cdot 2$
$=$ $8$
$(2 \sqrt{7} - 3) (1 - \sqrt{28})$

For this task you need the following basic knowledge: Powers
$(2 \sqrt{7} - 3) (1 - \sqrt{28})$ $=$
↓
Pull the 2 out of the root as: $2 \sqrt{7} = \sqrt{4 \cdot 7}$
$=$ $(\sqrt{28} - 3) (1 - \sqrt{28})$
↓
Multiply out the brackets.
$=$ $\sqrt{28} - 28 - 3 + 3 \sqrt{28}$
↓
Simplify.
$=$ $\sqrt{28} - 31 + 3 \sqrt{28}$
$=$ $4 \sqrt{28} - 31$
$3 \sqrt{63} + 6 \sqrt{72} - 4 \sqrt{28} - 17 \sqrt{8}$

For this task you need the following basic knowledge: Powers
$3 \sqrt{63} + 6 \sqrt{72} - 4 \sqrt{28} - 17 \sqrt{8}$ $=$
↓
Factorise the values under the root.
$=$ $3 \sqrt{7 \cdot 9} + 6 \sqrt{2 \cdot 36} - 4 \sqrt{7 \cdot 4} - 17 \sqrt{2 \cdot 4}$
↓
Take square roots.
$=$ $3 \cdot 3 \sqrt{7} + 6 \cdot 6 \sqrt{2} - 4 \cdot 2 \sqrt{7} - 34 \sqrt{2}$
$=$ $9 \sqrt{7} + 36 \sqrt{2} - 8 \sqrt{7} - 34 \sqrt{2}$
↓
Conclude.
$=$ $\sqrt{7} + 2 \sqrt{2}$

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