Question: 1
Simplify
7\sqrt{11} \times 3\sqrt{11}
2
Expand and simplify
(\sqrt{2} + 2)(\sqrt{2} + 5)
3
Rationalise the denominator
\frac{70}{\sqrt{10}}
4
Simplify
10\sqrt{7} + \sqrt{112}
1. **Simplify** $7\sqrt{11} \times 3\sqrt{11}$.
The problem asks us to multiply two terms involving square roots.
Formula: $a\sqrt{b} \times c\sqrt{b} = (a \times c) \times (\sqrt{b} \times \sqrt{b}) = ac \times b$
Step 1: Multiply the coefficients: $7 \times 3 = 21$
Step 2: Multiply the square roots: $\sqrt{11} \times \sqrt{11} = \sqrt{11 \times 11} = \sqrt{121} = 11$
Step 3: Combine: $21 \times 11 = 231$
**Answer:** $231$
2. **Expand and simplify** $(\sqrt{2} + 2)(\sqrt{2} + 5)$.
Use distributive property (FOIL): $(a+b)(c+d) = ac + ad + bc + bd$
Step 1: Multiply each term:
$\sqrt{2} \times \sqrt{2} = \sqrt{4} = 2$
$\sqrt{2} \times 5 = 5\sqrt{2}$
$2 \times \sqrt{2} = 2\sqrt{2}$
$2 \times 5 = 10$
Step 2: Add all terms:
$2 + 5\sqrt{2} + 2\sqrt{2} + 10$
Step 3: Combine like terms:
$5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$
$2 + 10 = 12$
**Answer:** $12 + 7\sqrt{2}$
3. **Rationalise the denominator** $\frac{70}{\sqrt{10}}$.
To rationalise, multiply numerator and denominator by $\sqrt{10}$:
$$\frac{70}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{70\sqrt{10}}{\sqrt{10} \times \sqrt{10}} = \frac{70\sqrt{10}}{10}$$
Step 2: Simplify fraction:
$$\frac{70\sqrt{10}}{\cancel{10}} = \frac{\cancel{70}7\sqrt{10}}{\cancel{10}1} = 7\sqrt{10}$$
**Answer:** $7\sqrt{10}$
4. **Simplify** $10\sqrt{7} + \sqrt{112}$.
Step 1: Simplify $\sqrt{112}$.
Factor 112: $112 = 16 \times 7$
$$\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16} \times \sqrt{7} = 4\sqrt{7}$$
Step 2: Rewrite expression:
$$10\sqrt{7} + 4\sqrt{7}$$
Step 3: Combine like terms:
$$10\sqrt{7} + 4\sqrt{7} = (10 + 4)\sqrt{7} = 14\sqrt{7}$$
**Answer:** $14\sqrt{7}$