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Exercises: Square Roots

Simplify these root terms as far as possible.

$\sqrt{20}$

$\sqrt{20}$ $=$
↓
Do a prime factorization.
$=$ $\sqrt{2 \cdot 2 \cdot 5}$
↓
Search for even exponents in the decomposition.
$=$ $\sqrt{2^{2} \cdot 5}$
↓
Decompose the square root.
$=$ $\sqrt{2^{2}} \cdot \sqrt{5}$
↓
Resolve the square root.
$=$ $2 \cdot \sqrt{5}$
$=$ $2 \sqrt{5}$
$\sqrt{27}$

$\sqrt{27}$ $=$
↓
Do a prime factorization.
$=$ $\sqrt{3 \cdot 3 \cdot 3}$
↓
Search for even exponents in the decomposition.
$=$ $\sqrt{3^{2} \cdot 3}$
↓
Decompose the square root.
$=$ $\sqrt{3^{2}} \cdot \sqrt{3}$
↓
Resolve the square root.
$=$ $3 \cdot \sqrt{3}$
$=$ $3 \sqrt{3}$
$\sqrt{45}$

$\sqrt{45}$ $=$
↓
Do a prime factorization.
$=$ $\sqrt{3 \cdot 3 \cdot 5}$
↓
Search for even exponents in the decomposition.
$=$ $\sqrt{3^{2} \cdot 5}$
↓
Decompose the square root.
$=$ $\sqrt{3^{2 \cdot 5}}$
↓
Resolve the square root.
$=$ $3 \cdot \sqrt{5}$
$=$ $3 \sqrt{5}$
$\sqrt{98}$

$\sqrt{98}$ $=$
↓
Do a prime factorization.
$=$ $\sqrt{2 \cdot 7 \cdot 7}$
↓
Search for even exponents in the decomposition.
$=$ $\sqrt{2 \cdot 7^{2}}$
↓
Decompose the square root.
$=$ $\sqrt{2} \cdot \sqrt{7^{2}}$
↓
Resolve the square root.
$=$ $\sqrt{2} \cdot 7$
$=$ $7 \sqrt{2}$

$\sqrt{180}$

$\frac{\sqrt{8}}{2}$

$\frac{\sqrt{8}}{2}$ $=$
↓
Do a prime factorization.
$=$ $\frac{\sqrt{2 \cdot 2 \cdot 2}}{2}$
↓
Search for even exponents in the decomposition.
$=$ $\frac{\sqrt{2^{2} \cdot 2}}{2}$
↓
Decompose the square root.
$=$ $\frac{\sqrt{2^{2}} \cdot \sqrt{2}}{2}$
↓
Resolve the square root.
$=$ $\frac{2 \cdot \sqrt{2}}{2}$
$=$ $\sqrt{2}$

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