Question

lesson 4: to be determined
ready, set, go
ready
rewrite as multiples of \\(\sqrt{2}\\).

1. \\(\sqrt{8}\\)
2. \\(\sqrt{18}\\)
3. \\(\sqrt{32}\\)

rewrite as multiples of \\(\sqrt{5}\\).

1. \\(\sqrt{20}\\)
2. \\(\sqrt{45}\\)
3. \\(\sqrt{80}\\)
4. explain how rewriting radical expressions makes it possible to add or subtract them.

Explanation:

Step1: Factor radicand for $\sqrt{8}$

$\sqrt{8}=\sqrt{4\times2}$

Step2: Split square root

$\sqrt{4\times2}=\sqrt{4}\times\sqrt{2}$

Step3: Simplify $\sqrt{4}$

$\sqrt{4}\times\sqrt{2}=2\sqrt{2}$

Step1: Factor radicand for $\sqrt{18}$

$\sqrt{18}=\sqrt{9\times2}$

Step2: Split square root

$\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}$

Step3: Simplify $\sqrt{9}$

$\sqrt{9}\times\sqrt{2}=3\sqrt{2}$

Step1: Factor radicand for $\sqrt{32}$

$\sqrt{32}=\sqrt{16\times2}$

Step2: Split square root

$\sqrt{16\times2}=\sqrt{16}\times\sqrt{2}$

Step3: Simplify $\sqrt{16}$

$\sqrt{16}\times\sqrt{2}=4\sqrt{2}$

Step1: Factor radicand for $\sqrt{20}$

$\sqrt{20}=\sqrt{4\times5}$

Step2: Split square root

$\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}$

Step3: Simplify $\sqrt{4}$

$\sqrt{4}\times\sqrt{5}=2\sqrt{5}$

Step1: Factor radicand for $\sqrt{45}$

$\sqrt{45}=\sqrt{9\times5}$

Step2: Split square root

$\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}$

Step3: Simplify $\sqrt{9}$

$\sqrt{9}\times\sqrt{5}=3\sqrt{5}$

Step1: Factor radicand for $\sqrt{80}$

$\sqrt{80}=\sqrt{16\times5}$

Step2: Split square root

$\sqrt{16\times5}=\sqrt{16}\times\sqrt{5}$

Step3: Simplify $\sqrt{16}$

$\sqrt{16}\times\sqrt{5}=4\sqrt{5}$

Step1: Explain radical addition/subtraction rule

Radicals can only be combined if they have identical radicands (like radicals). Rewriting radicals simplifies them to a form where the radicands match, allowing us to add or subtract the coefficients in front of the shared radical term, just like combining like terms such as $2x + 3x = 5x$.

Answer:

1. $2\sqrt{2}$
2. $3\sqrt{2}$
3. $4\sqrt{2}$
4. $2\sqrt{5}$
5. $3\sqrt{5}$
6. $4\sqrt{5}$
7. Rewriting radicals creates like radicals, which can be combined by adding/subtracting their coefficients, similar to combining like algebraic terms.