Question

there are some radical expressions: \\(\sqrt{135} - \sqrt{60}\\), \\(\sqrt{72} + \sqrt{8}\\), \\(\sqrt{7} \cdot \sqrt{9}\\), \\(3\sqrt{7}\\), \\(1\sqrt{15}\\), \\(8\sqrt{2}\\) (the image shows a matching or calculation - related layout with these radical expressions).

Explanation:

To solve the problems involving square roots, we will simplify each expression step by step:

1. Simplify $\boldsymbol{\sqrt{135} - \sqrt{60}}$

Step 1: Factor the radicands

Factor 135 and 60 to find perfect square factors.

• $135 = 9 \times 15$, so $\sqrt{135} = \sqrt{9 \times 15} = \sqrt{9} \times \sqrt{15} = 3\sqrt{15}$
• $60 = 4 \times 15$, so $\sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15} = 2\sqrt{15}$

Step 2: Subtract the simplified radicals

Now subtract the two simplified radicals:
$3\sqrt{15} - 2\sqrt{15} = (3 - 2)\sqrt{15} = \sqrt{15}$

2. Simplify $\boldsymbol{\sqrt{72} + \sqrt{8}}$

Step 1: Factor the radicands

Factor 72 and 8 to find perfect square factors.

• $72 = 36 \times 2$, so $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$
• $8 = 4 \times 2$, so $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

Step 2: Add the simplified radicals

Now add the two simplified radicals:
$6\sqrt{2} + 2\sqrt{2} = (6 + 2)\sqrt{2} = 8\sqrt{2}$

3. Simplify $\boldsymbol{\sqrt{7} \cdot \sqrt{9}}$

Step 1: Use the property of square roots

Recall that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ (for $a \geq 0, b \geq 0$). Also, $\sqrt{9} = 3$.
So, $\sqrt{7} \cdot \sqrt{9} = \sqrt{7} \cdot 3 = 3\sqrt{7}$

Final Answers:

• $\sqrt{135} - \sqrt{60} = \boldsymbol{\sqrt{15}}$
• $\sqrt{72} + \sqrt{8} = \boldsymbol{8\sqrt{2}}$
• $\sqrt{7} \cdot \sqrt{9} = \boldsymbol{3\sqrt{7}}$

Answer:

To solve the problems involving square roots, we will simplify each expression step by step:

1. Simplify $\boldsymbol{\sqrt{135} - \sqrt{60}}$

Step 1: Factor the radicands

Factor 135 and 60 to find perfect square factors.

• $135 = 9 \times 15$, so $\sqrt{135} = \sqrt{9 \times 15} = \sqrt{9} \times \sqrt{15} = 3\sqrt{15}$
• $60 = 4 \times 15$, so $\sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15} = 2\sqrt{15}$

Step 2: Subtract the simplified radicals

Now subtract the two simplified radicals:
$3\sqrt{15} - 2\sqrt{15} = (3 - 2)\sqrt{15} = \sqrt{15}$

2. Simplify $\boldsymbol{\sqrt{72} + \sqrt{8}}$

Step 1: Factor the radicands

Factor 72 and 8 to find perfect square factors.

• $72 = 36 \times 2$, so $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$
• $8 = 4 \times 2$, so $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

Step 2: Add the simplified radicals

Now add the two simplified radicals:
$6\sqrt{2} + 2\sqrt{2} = (6 + 2)\sqrt{2} = 8\sqrt{2}$

3. Simplify $\boldsymbol{\sqrt{7} \cdot \sqrt{9}}$

Step 1: Use the property of square roots

Recall that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ (for $a \geq 0, b \geq 0$). Also, $\sqrt{9} = 3$.
So, $\sqrt{7} \cdot \sqrt{9} = \sqrt{7} \cdot 3 = 3\sqrt{7}$

Final Answers:

• $\sqrt{135} - \sqrt{60} = \boldsymbol{\sqrt{15}}$
• $\sqrt{72} + \sqrt{8} = \boldsymbol{8\sqrt{2}}$
• $\sqrt{7} \cdot \sqrt{9} = \boldsymbol{3\sqrt{7}}$