Question

1. \\(\sqrt{1000}\\)\
2. \\(\sqrt{32}\\)\
3. \\(\sqrt{12} \cdot \sqrt{3}\\)

Explanation:

Problem 12: Simplify $\boldsymbol{\sqrt{1000}}$

Step1: Factor 1000

Factor 1000 into a perfect square and another number: $1000 = 100 \times 10$.

Step2: Use square root property

Use $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$: $\sqrt{1000}=\sqrt{100\times 10}=\sqrt{100}\cdot\sqrt{10}$.

Step3: Simplify $\sqrt{100}$

$\sqrt{100} = 10$, so $\sqrt{1000}=10\sqrt{10}$.

Step1: Factor 32

Factor 32 into a perfect square and another number: $32 = 16 \times 2$.

Step2: Use square root property

Use $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$: $\sqrt{32}=\sqrt{16\times 2}=\sqrt{16}\cdot\sqrt{2}$.

Step3: Simplify $\sqrt{16}$

$\sqrt{16} = 4$, so $\sqrt{32}=4\sqrt{2}$.

Step1: Use square root multiplication property

Use $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$: $\sqrt{12}\cdot\sqrt{3}=\sqrt{12\times 3}$.

Step2: Multiply inside the square root

$12\times 3 = 36$, so $\sqrt{12\times 3}=\sqrt{36}$.

Step3: Simplify $\sqrt{36}$

$\sqrt{36} = 6$.

Answer:

$10\sqrt{10}$

Problem 15: Simplify $\boldsymbol{\sqrt{32}}$