practice simplify each radical expression. 1. $3\sqrt{5} \cdot 2\sqrt{5}$ 2. $4\sqrt{80}$ 3. $\sqrt{3} \cdot \sqrt{36}$ 4. $\sqrt{18}$ 5. $\sqrt{63}$ 6. $2\sqrt{28}$ 7. $\sqrt{25y^8}$ 8. $6\sqrt{48b^7}$ 9. $\sqrt{8} \cdot \sqrt{32x^9}$ skills review & practice • ns10 copyright © savvas learning company llc. all rights

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Accepted Answer

Let's solve each radical expression one by one:

### 1. $ 3\sqrt{5} \cdot 2\sqrt{5} $
## Step 1: Multiply the coefficients and the radicals separately
Multiply the coefficients $ 3 $ and $ 2 $, and multiply the radicals $ \sqrt{5} $ and $ \sqrt{5} $.
$ (3 \cdot 2) \cdot (\sqrt{5} \cdot \sqrt{5}) $
## Step 2: Simplify
$ 3 \cdot 2 = 6 $ and $ \sqrt{5} \cdot \sqrt{5} = \sqrt{5^2} = 5 $
So, $ 6 \cdot 5 = 30 $
# Answer: $ 30 $

### 2. $ 4\sqrt{80} $
## Step 1: Factor 80 into perfect square and other factor
$ 80 = 16 \times 5 $, where $ 16 $ is a perfect square.
$ 4\sqrt{16 \times 5} $
## Step 2: Use the property $ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} $
$ 4 \cdot \sqrt{16} \cdot \sqrt{5} $
## Step 3: Simplify $ \sqrt{16} $
$ \sqrt{16} = 4 $
So, $ 4 \cdot 4 \cdot \sqrt{5} = 16\sqrt{5} $
# Answer: $ 16\sqrt{5} $

### 3. $ \sqrt{3} \cdot \sqrt{36} $
## Step 1: Simplify $ \sqrt{36} $
$ \sqrt{36} = 6 $
## Step 2: Multiply with $ \sqrt{3} $
$ \sqrt{3} \cdot 6 = 6\sqrt{3} $
# Answer: $ 6\sqrt{3} $

### 4. $ \sqrt{18} $
## Step 1: Factor 18 into perfect square and other factor
$ 18 = 9 \times 2 $, where $ 9 $ is a perfect square.
$ \sqrt{9 \times 2} $
## Step 2: Use the property $ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} $
$ \sqrt{9} \cdot \sqrt{2} $
## Step 3: Simplify $ \sqrt{9} $
$ \sqrt{9} = 3 $
So, $ 3\sqrt{2} $
# Answer: $ 3\sqrt{2} $

### 5. $ \sqrt{63} $
## Step 1: Factor 63 into perfect square and other factor
$ 63 = 9 \times 7 $, where $ 9 $ is a perfect square.
$ \sqrt{9 \times 7} $
## Step 2: Use the property $ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} $
$ \sqrt{9} \cdot \sqrt{7} $
## Step 3: Simplify $ \sqrt{9} $
$ \sqrt{9} = 3 $
So, $ 3\sqrt{7} $
# Answer: $ 3\sqrt{7} $

### 6. $ 2\sqrt{28} $
## Step 1: Factor 28 into perfect square and other factor
$ 28 = 4 \times 7 $, where $ 4 $ is a perfect square.
$ 2\sqrt{4 \times 7} $
## Step 2: Use the property $ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} $
$ 2 \cdot \sqrt{4} \cdot \sqrt{7} $
## Step 3: Simplify $ \sqrt{4} $
$ \sqrt{4} = 2 $
So, $ 2 \cdot 2 \cdot \sqrt{7} = 4\sqrt{7} $
# Answer: $ 4\sqrt{7} $

### 7. $ \sqrt{25y^8} $
## Step 1: Use the property $ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} $
$ \sqrt{25} \cdot \sqrt{y^8} $
## Step 2: Simplify $ \sqrt{25} $ and $ \sqrt{y^8} $
$ \sqrt{25} = 5 $ and $ \sqrt{y^8} = y^4 $ (since $ (y^4)^2 = y^8 $)
So, $ 5 \cdot y^4 = 5y^4 $
# Answer: $ 5y^4 $

### 8. $ 6\sqrt{48b^7} $
## Step 1: Factor 48 and $ b^7 $ into perfect square and other factor
$ 48 = 16 \times 3 $ and $ b^7 = b^6 \times b $, where $ 16 $ and $ b^6 $ are perfect squares.
$ 6\sqrt{16 \times 3 \times b^6 \times b} $
## Step 2: Use the property $ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} $
$ 6 \cdot \sqrt{16} \cdot \sqrt{3} \cdot \sqrt{b^6} \cdot \sqrt{b} $
## Step 3: Simplify the perfect squares
$ \sqrt{16} = 4 $ and $ \sqrt{b^6} = b^3 $ (since $ (b^3)^2 = b^6 $)
So, $ 6 \cdot 4 \cdot b^3 \cdot \sqrt{3b} = 24b^3\sqrt{3b} $
# Answer: $ 24b^3\sqrt{3b} $

### 9. $ \sqrt{8} \cdot \sqrt{32x^9} $
## Step 1: Use the property $ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} $
$ \sqrt{8 \times 32x^9} $
## Step 2: Multiply 8 and 32
$ 8 \times 32 = 256 $
So, $ \sqrt{256x^9} $
## Step 3: Factor $ x^9 $ into perfect square and other factor
$ x^9 = x^8 \times x $, where $ x^8 $ is a perfect square.
$ \sqrt{256 \times x^8 \times x} $
## Step 4: Use the property $ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} $
$ \sqrt{256} \cdot \sqrt{x^8} \cdot \sqrt{x} $
## Step 5: Simplify the perfect squares
$ \sqrt{256} = 16 $ and $ \sqrt{x^8} = x^4 $ (since $ (x^4)^2 = x^8 $)
So, $ 16 \cdot x^4 \cdot \sqrt{x} = 16x^4\sqrt{x} $
# Answer: $ 16x^4\sqrt{x} $

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QUESTION IMAGE

Question

Explanation:

1. \( 3\sqrt{5} \cdot 2\sqrt{5} \)

Step 1: Multiply the coefficients and the radicals separately

Step 2: Simplify

Answer:

2. \( 4\sqrt{80} \)

Step 1: Factor 80 into perfect square and other factor

Step 2: Use the property \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \)

Step 3: Simplify \( \sqrt{16} \)