unit 3, lesson 4 – ready, set, go
ready
rewrite as multiples of $\sqrt{2}$.
1. $\sqrt{8}$ 2. $\sqrt{18}$ 3. $\sqrt{32}$
rewrite as multiples of $\sqrt{5}$.
4. $\sqrt{20}$ 5. $\sqrt{45}$ 6. $\sqrt{80}$
7. explain how rewriting radical expressions makes it possible to add or subtract them.

Question

Accepted Answer

### Problem 1: $\boldsymbol{\sqrt{8}}$
# Explanation:
## Step1: Factor 8 into 4×2
$\sqrt{8} = \sqrt{4	imes2}$
## Step2: Use property $\sqrt{ab}=\sqrt{a}	imes\sqrt{b}$
$\sqrt{4	imes2} = \sqrt{4}	imes\sqrt{2}$
## Step3: Simplify $\sqrt{4}$
$\sqrt{4} = 2$, so $\sqrt{8} = 2\sqrt{2}$
# Answer: $2\sqrt{2}$

### Problem 2: $\boldsymbol{\sqrt{18}}$
# Explanation:
## Step1: Factor 18 into 9×2
$\sqrt{18} = \sqrt{9	imes2}$
## Step2: Use property $\sqrt{ab}=\sqrt{a}	imes\sqrt{b}$
$\sqrt{9	imes2} = \sqrt{9}	imes\sqrt{2}$
## Step3: Simplify $\sqrt{9}$
$\sqrt{9} = 3$, so $\sqrt{18} = 3\sqrt{2}$
# Answer: $3\sqrt{2}$

### Problem 3: $\boldsymbol{\sqrt{32}}$
# Explanation:
## Step1: Factor 32 into 16×2
$\sqrt{32} = \sqrt{16	imes2}$
## Step2: Use property $\sqrt{ab}=\sqrt{a}	imes\sqrt{b}$
$\sqrt{16	imes2} = \sqrt{16}	imes\sqrt{2}$
## Step3: Simplify $\sqrt{16}$
$\sqrt{16} = 4$, so $\sqrt{32} = 4\sqrt{2}$
# Answer: $4\sqrt{2}$

### Problem 4: $\boldsymbol{\sqrt{20}}$
# Explanation:
## Step1: Factor 20 into 4×5
$\sqrt{20} = \sqrt{4	imes5}$
## Step2: Use property $\sqrt{ab}=\sqrt{a}	imes\sqrt{b}$
$\sqrt{4	imes5} = \sqrt{4}	imes\sqrt{5}$
## Step3: Simplify $\sqrt{4}$
$\sqrt{4} = 2$, so $\sqrt{20} = 2\sqrt{5}$
# Answer: $2\sqrt{5}$

### Problem 5: $\boldsymbol{\sqrt{45}}$
# Explanation:
## Step1: Factor 45 into 9×5
$\sqrt{45} = \sqrt{9	imes5}$
## Step2: Use property $\sqrt{ab}=\sqrt{a}	imes\sqrt{b}$
$\sqrt{9	imes5} = \sqrt{9}	imes\sqrt{5}$
## Step3: Simplify $\sqrt{9}$
$\sqrt{9} = 3$, so $\sqrt{45} = 3\sqrt{5}$
# Answer: $3\sqrt{5}$

### Problem 6: $\boldsymbol{\sqrt{80}}$
# Explanation:
## Step1: Factor 80 into 16×5
$\sqrt{80} = \sqrt{16	imes5}$
## Step2: Use property $\sqrt{ab}=\sqrt{a}	imes\sqrt{b}$
$\sqrt{16	imes5} = \sqrt{16}	imes\sqrt{5}$
## Step3: Simplify $\sqrt{16}$
$\sqrt{16} = 4$, so $\sqrt{80} = 4\sqrt{5}$
# Answer: $4\sqrt{5}$

### Problem 7: Explain...
# Brief Explanations:
To add or subtract radical expressions, they must have the same radical part (like radicals). Rewriting radicals (simplifying) involves factoring out perfect squares from the radicand to express the radical in simplest form (e.g., $\sqrt{8}=2\sqrt{2}$). This process reveals the simplest radical form, allowing us to identify if the radical parts are the same. If they are, we can combine the coefficients (e.g., $2\sqrt{2} + 3\sqrt{2} = (2 + 3)\sqrt{2} = 5\sqrt{2}$). Without simplifying, we might not recognize like radicals (e.g., $\sqrt{8}$ and $\sqrt{18}$ look different initially but simplify to $2\sqrt{2}$ and $3\sqrt{2}$, which are like radicals and can be added/subtracted).
# Answer: Rewriting radical expressions simplifies them to have the same radical part (like radicals) when possible. Only like radicals (same radicand and index) can be added/subtracted by combining their coefficients. Simplifying reveals if radicals are like, enabling addition/subtraction (e.g., $\sqrt{8}=2\sqrt{2}$, $\sqrt{18}=3\sqrt{2}$; $2\sqrt{2}+3\sqrt{2}=5\sqrt{2}$).

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QUESTION IMAGE

Question

Explanation:

Problem 1: $\boldsymbol{\sqrt{8}}$

Step1: Factor 8 into 4×2

Step2: Use property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$

Step3: Simplify $\sqrt{4}$

Step1: Factor 18 into 9×2

Step2: Use property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$

Step3: Simplify $\sqrt{9}$

Step1: Factor 32 into 16×2

Step2: Use property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$

Step3: Simplify $\sqrt{16}$

Answer:

Problem 2: $\boldsymbol{\sqrt{18}}$