https://r2.erweima.ai/mathbot/image/17249235962475shm6g8u.pdf

please provide step-by-step solutions for all the questions

Here are step-by-step solutions for the questions in the PDF:

### **Year 11 into 12 Task 1 - Indices & Surds**

#### **Question 1:**
Simplify the ratio \(\sqrt{12} : \sqrt{48} : \sqrt{300}\).

**Step 1:** Simplify each square root individually.
- \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)
- \(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)
- \(\sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3}\)

**Step 2:** Express the ratio in its simplest form.
\[
\text{Ratio} = 2\sqrt{3} : 4\sqrt{3} : 10\sqrt{3}
\]
Divide all terms by \(\sqrt{3}\):
\[
\text{Simplified Ratio} = 2 : 4 : 10
\]
Further divide by 2:
\[
\text{Final Ratio} = 1 : 2 : 5
\]

---

#### **Question 2:**
Write \(\sqrt{75} + \sqrt{12}\) in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers.

**Step 1:** Simplify each square root.
- \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\)
- \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)

**Step 2:** Add the terms.
\[
\sqrt{75} + \sqrt{12} = 5\sqrt{3} + 2\sqrt{3} = (5 + 2)\sqrt{3} = 7\sqrt{3}
\]

So, the answer is \(7\sqrt{3}\).

---

#### **Question 3:**
Write \(\sqrt{500} - 2\sqrt{45}\) in the form \(a\sqrt{5}\) where \(a\) is an integer.

**Step 1:** Simplify each square root.
- \(\sqrt{500} = \sqrt{100 \times 5} = 10\sqrt{5}\)
- \(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\)

**Step 2:** Multiply the second term by 2:
\[
2\sqrt{45} = 2 \times 3\sqrt{5} = 6\sqrt{5}
\]

**Step 3:** Subtract the terms.
\[
\sqrt{500} - 2\sqrt{45} = 10\sqrt{5} - 6\sqrt{5} = (10 - 6)\sqrt{5} = 4\sqrt{5}
\]

So, the answer is \(4\sqrt{5}\).

---

#### **Question 4:**
Expand and simplify \((\sqrt{5} + 3)(\sqrt{5} - 2)(\sqrt{5} + 1)\).

**Step 1:** Expand the first two terms.
\[
(\sqrt{5} + 3)(\sqrt{5} - 2) = \sqrt{5}\sqrt{5} - 2\sqrt{5} + 3\sqrt{5} - 6 = 5 + \sqrt{5} - 6 = \sqrt{5} - 1
\]

**Step 2:** Multiply the result by \(\sqrt{5} + 1\).
\[
(\sqrt{5} - 1)(\sqrt{5} + 1) = \sqrt{5}\sqrt{5} + \sqrt{5} - \sqrt{5} - 1 = 5 - 1 = 4
\]

So, the answer is \(4\).

---

#### **Question 5:**
Rationalize the denominator of \(\frac{8}{3 - \sqrt{5}}\). Give your answer in the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are integers.

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator.
\[
\frac{8}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} = \frac{8(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}
\]

**Step 2:** Simplify the denominator.
\[
(3 - \sqrt{5})(3 + \sqrt{5}) = 9 - 5 = 4
\]

**Step 3:** Simplify the numerator.
\[
\frac{8(3 + \sqrt{5})}{4} = \frac{24 + 8\sqrt{5}}{4} = 6 + 2\sqrt{5}
\]

So, the answer is \(6 + 2\sqrt{5}\).

---

Would you like to go through more detailed solutions or continue with other questions? Here are 5 questions for further exploration:

1. How do you simplify the expression involving indices and surds?
2. What is the process for rationalizing denominators with surds?
3. How can expanding and simplifying products of surds lead to simpler expressions?
4. What strategies can be used to solve equations involving square roots?
5. How do you find the exact values of trigonometric functions using the cosine rule?

**Tip:** Always verify your work by substituting your answers back into the original equation or expression to ensure accuracy.

Explore step-by-step solutions for Year 11-12 mathematical questions on indices, surds, and rationalizing denominators. Simplify ratios, add and subtract square roots, expand and simplify products of surds, and rationalize denominators. Perfect for Year 11-12 students.

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