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Exercises: Terms with Square Roots

Simplify the following terms.

$\sqrt{500} + 3 \sqrt{98} - 5 \sqrt{8} - 3 \sqrt{45}$

For this task you need the following basic knowledge: Square Roots
$=$ $\sqrt{500} + 3 \sqrt{98} - 5 \sqrt{8} - 3 \sqrt{45}$
↓
Decompose the numbers under the roots.
$=$ $\sqrt{5 \cdot 10^{2}} + 3 \sqrt{2 \cdot 7^{2}} - 5 \sqrt{2^{2} \cdot 2} - 3 \sqrt{3^{2} \cdot 5}$
↓
Pull the powers out of the root.
$=$ $10 \sqrt{5} + 3 \cdot 7 \sqrt{2} - 5 \cdot 2 \sqrt{2} - 3 \cdot 3 \sqrt{5}$
$=$ $10 \sqrt{5} + 21 \sqrt{2} - 10 \sqrt{2} - 9 \sqrt{5}$
$=$ $\sqrt{5} + 11 \sqrt{2}$
$\sqrt{64 k^{2}}$

For this task you need the following basic knowledge: Square Roots
$=$ $\sqrt{64 k^{2}}$
↓
Decompose the numbers under the root.
$=$ $\sqrt{8^{2} \cdot k^{2}}$
↓
Pull the powers out of the root.
(Don't forget the absolute value around k.)
$=$ $8 \cdot | k |$
$(\sqrt{\frac{x^{5} y}{5 a}} : \sqrt{\frac{x^{3} y^{3}}{a^{2}}}) \cdot \sqrt{\frac{25 x}{a}} (x, y, z > 0)$

For this task you need the following basic knowledge: Square Roots
$=$ $(\sqrt{\frac{x^{5} y}{5 a}} : \sqrt{\frac{x^{3} y^{3}}{a^{2}}}) \cdot \sqrt{\frac{25 x}{a}}$
↓
Conclude the fractions.
$=$ $\frac{\sqrt{x^{5} y} \cdot \sqrt{a^{2}}}{\sqrt{5 a} \cdot \sqrt{x^{3} y^{3}}} \cdot \frac{\sqrt{25 x}}{\sqrt{a}}$
$=$ $\sqrt{\frac{x^{5} y \cdot a^{2} \cdot 25 x}{5 a x^{3} y^{3} a}}$
↓
Shorten the fraction.
$=$ $\sqrt{\frac{5 x^{3}}{y^{2}}}$
↓
Pull the powers out of the root.
$=$ $\frac{x}{y} \sqrt{5 x}$

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