13) $sqrt{15}(sqrt{3} + 2)$
14) $sqrt{6}(sqrt{2} + sqrt{3})$
15) $sqrt{10}(sqrt{10} + 2)$
16) $sqrt{5}(sqrt{5} + 3)$
17) $sqrt{5}(5 + sqrt{5})$
18) $sqrt{15}(sqrt{3} + sqrt{10})$
19) $-5sqrt{3}(2 + sqrt{5})$
20) $sqrt{3}(5 + sqrt{2})$
21) $sqrt{10}(4sqrt{2} + sqrt{5})$
22) $3sqrt{6}(sqrt{10} - sqrt{3})$
23) $-4sqrt{5}(4 - 3sqrt{10})$
24) $sqrt{5}(-4sqrt{6} + sqrt{10})$

Question

Accepted Answer

Let's solve each problem one by one using the distributive property (also known as the distributive law of multiplication over addition/subtraction), which states that $a(b + c)=ab+ac$.

### Problem 13: $\boldsymbol{\sqrt{15}(\sqrt{3}+2)}$
## Step1: Apply the distributive property
$\sqrt{15}\times\sqrt{3}+\sqrt{15}\times2$
We know that $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$, so $\sqrt{15}\times\sqrt{3}=\sqrt{15\times3}=\sqrt{45}=\sqrt{9\times5} = 3\sqrt{5}$ and $\sqrt{15}\times2 = 2\sqrt{15}$
## Step2: Combine the terms
$3\sqrt{5}+2\sqrt{15}$
# Answer: $3\sqrt{5}+2\sqrt{15}$

### Problem 14: $\boldsymbol{\sqrt{6}(\sqrt{2}+\sqrt{3})}$
## Step1: Apply the distributive property
$\sqrt{6}\times\sqrt{2}+\sqrt{6}\times\sqrt{3}$
Using $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$, we have $\sqrt{6}\times\sqrt{2}=\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}$ and $\sqrt{6}\times\sqrt{3}=\sqrt{18}=\sqrt{9\times2} = 3\sqrt{2}$
## Step2: Combine the terms
$2\sqrt{3}+3\sqrt{2}$
# Answer: $2\sqrt{3}+3\sqrt{2}$

### Problem 15: $\boldsymbol{\sqrt{10}(\sqrt{10}+2)}$
## Step1: Apply the distributive property
$\sqrt{10}\times\sqrt{10}+\sqrt{10}\times2$
We know that $\sqrt{a}\times\sqrt{a}=a$, so $\sqrt{10}\times\sqrt{10} = 10$ and $\sqrt{10}\times2=2\sqrt{10}$
## Step2: Combine the terms
$10 + 2\sqrt{10}$
# Answer: $10 + 2\sqrt{10}$

### Problem 16: $\boldsymbol{\sqrt{5}(\sqrt{5}+3)}$
## Step1: Apply the distributive property
$\sqrt{5}\times\sqrt{5}+\sqrt{5}\times3$
Since $\sqrt{5}\times\sqrt{5}=5$ and $\sqrt{5}\times3 = 3\sqrt{5}$
## Step2: Combine the terms
$5+3\sqrt{5}$
# Answer: $5 + 3\sqrt{5}$

### Problem 17: $\boldsymbol{\sqrt{5}(5+\sqrt{5})}$
## Step1: Apply the distributive property
$\sqrt{5}\times5+\sqrt{5}\times\sqrt{5}$
We get $5\sqrt{5}+5$ (because $\sqrt{5}\times\sqrt{5} = 5$)
## Step2: Rearrange the terms
$5 + 5\sqrt{5}$
# Answer: $5+5\sqrt{5}$

### Problem 18: $\boldsymbol{\sqrt{15}(\sqrt{3}+\sqrt{10})}$
## Step1: Apply the distributive property
$\sqrt{15}\times\sqrt{3}+\sqrt{15}\times\sqrt{10}$
Using $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$, $\sqrt{15}\times\sqrt{3}=\sqrt{45}=3\sqrt{5}$ and $\sqrt{15}\times\sqrt{10}=\sqrt{150}=\sqrt{25\times6}=5\sqrt{6}$
## Step2: Combine the terms
$3\sqrt{5}+5\sqrt{6}$
# Answer: $3\sqrt{5}+5\sqrt{6}$

### Problem 19: $\boldsymbol{- 5\sqrt{3}(2+\sqrt{5})}$
## Step1: Apply the distributive property
$-5\sqrt{3}\times2+(-5\sqrt{3})\times\sqrt{5}$
We have $-10\sqrt{3}-5\sqrt{15}$ (since $\sqrt{3}\times\sqrt{5}=\sqrt{15}$)
## Step2: Combine the terms
$-10\sqrt{3}-5\sqrt{15}$
# Answer: $-10\sqrt{3}-5\sqrt{15}$

### Problem 20: $\boldsymbol{\sqrt{3}(5+\sqrt{2})}$
## Step1: Apply the distributive property
$\sqrt{3}\times5+\sqrt{3}\times\sqrt{2}$
We get $5\sqrt{3}+\sqrt{6}$
## Step2: Combine the terms
$5\sqrt{3}+\sqrt{6}$
# Answer: $5\sqrt{3}+\sqrt{6}$

### Problem 21: $\boldsymbol{\sqrt{10}(4\sqrt{2}+\sqrt{5})}$
## Step1: Apply the distributive property
$\sqrt{10}\times4\sqrt{2}+\sqrt{10}\times\sqrt{5}$
Using $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$, $\sqrt{10}\times4\sqrt{2}=4\sqrt{20}=4\sqrt{4\times5}=8\sqrt{5}$ and $\sqrt{10}\times\sqrt{5}=\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}$
## Step2: Combine the terms
$8\sqrt{5}+5\sqrt{2}$
# Answer: $8\sqrt{5}+5\sqrt{2}$

### Problem 22: $\boldsymbol{3\sqrt{6}(\sqrt{10}-\sqrt{3})}$
## Step1: Apply the distributive property
$3\sqrt{6}\times\sqrt{10}-3\sqrt{6}\times\sqrt{3}$
We know that $\sqrt{6}\times\sqrt{10}=\sqrt{60}=\sqrt{4\times15}=2\sqrt{15}$ and $\sqrt{6}\times\sqrt{3}=\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}$
So $3\times2\sqrt{15}-3\times3\sqrt{2}=6\sqrt{15}-9\sqrt{2}$
## Step2: Combine the terms
$6\sqrt{15}-9\sqrt{2}$
# Answer: $6\sqrt{15}-9\sqrt{2}$

### Problem 23: $\boldsymbol{-4\sqrt{5}(4 - 3\sqrt{10})}$
## Step1: Apply the distributive property
$-4\sqrt{5}\times4-(-4\sqrt{5})\times3\sqrt{10}$
We have $-16\sqrt{5}+12\sqrt{50}$ (because $-4\sqrt{5}\times(- 3\sqrt{10})=12\sqrt{50}$)
Simplify $\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}$, so $12\sqrt{50}=12\times5\sqrt{2}=60\sqrt{2}$
## Step2: Combine the terms
$-16\sqrt{5}+60\sqrt{2}$
# Answer: $-16\sqrt{5}+60\sqrt{2}$

### Problem 24: $\boldsymbol{\sqrt{5}(-4\sqrt{6}+\sqrt{10})}$
## Step1: Apply the distributive property
$\sqrt{5}\times(-4\sqrt{6})+\sqrt{5}\times\sqrt{10}$
We get $-4\sqrt{30}+\sqrt{50}$ (since $\sqrt{5}\times\sqrt{6}=\sqrt{30}$ and $\sqrt{5}\times\sqrt{10}=\sqrt{50}$)
Simplify $\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}$
## Step2: Combine the terms
$-4\sqrt{30}+5\sqrt{2}$
# Answer: $-4\sqrt{30}+5\sqrt{2}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 13: \(\boldsymbol{\sqrt{15}(\sqrt{3}+2)}\)

Step1: Apply the distributive property

Step2: Combine the terms

Answer:

Problem 14: \(\boldsymbol{\sqrt{6}(\sqrt{2}+\sqrt{3})}\)

Step1: Apply the distributive property

Step2: Combine the terms