Let's solve the problem step by step.

Part (a) Simplify $\sqrt{x^{8} y^{10}} \div \sqrt[3]{x^{3} y^{-6}}$

We need to simplify the expression $\sqrt{x^{8} y^{10}} \div \sqrt[3]{x^{3} y^{-6}}$.

Step 1: Simplify each radical expression

1. $\sqrt{x^8 y^{10}}$: $\sqrt{x^8 y^{10}} = x^{\frac{8}{2}} y^{\frac{10}{2}} = x^4 y^5$

2. $\sqrt[3]{x^3 y^{-6}}$: $\sqrt[3]{x^3 y^{-6}} = x^{\frac{3}{3}} y^{\frac{-6}{3}} = x^1 y^{-2}$

Step 2: Divide the simplified expressions

$\frac{x^4 y^5}{x^1 y^{-2}} = x^{4-1} y^{5 - (-2)} = x^3 y^{7}$

Thus, the simplified form is $x^3 y^7$.

Part (b)(i) Show that $4(t-2)^{\frac{1}{2}}+5(t-2)^{\frac{3}{2}}$ can be written in the form $(t-2)^{p}(qt+r)$.

We want to express $4(t-2)^{\frac{1}{2}} + 5(t-2)^{\frac{3}{2}}$ in the form $(t-2)^p(qt + r)$.

Step 1: Factor out the common term $(t-2)^{\frac{1}{2}}$:

$4(t-2)^{\frac{1}{2}} + 5(t-2)^{\frac{3}{2}} = (t-2)^{\frac{1}{2}} \left(4 + 5(t-2)\right)$

Step 2: Simplify the expression inside the parentheses:

$4 + 5(t-2) = 4 + 5t - 10 = 5t - 6$

Thus, the expression can be written as: $4(t-2)^{\frac{1}{2}} + 5(t-2)^{\frac{3}{2}} = (t-2)^{\frac{1}{2}}(5t - 6)$

Here, $p = \frac{1}{2}$, $q = 5$, and $r = -6$.

Part (b)(ii) Solve the equation $4(t-2)^{\frac{1}{2}} + 5(t-2)^{\frac{3}{2}} = 0$.

From part (b)(i), we can rewrite the equation as: $(t-2)^{\frac{1}{2}}(5t - 6) = 0$

This equation is a product of two factors, so each factor can be set to zero.

Case 1: $(t-2)^{\frac{1}{2}} = 0$

$(t-2)^{\frac{1}{2}} = 0 \implies t-2 = 0 \implies t = 2$

Case 2: $5t - 6 = 0$

$5t - 6 = 0 \implies 5t = 6 \implies t = \frac{6}{5}$

Final Answer:

The solutions to the equation are $t = 2$ and $t = \frac{6}{5}$.

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Do you want further details or have any questions?

Here are some related questions for practice:

1. Simplify $\sqrt{x^{12} y^{8}} \div \sqrt[4]{x^{8} y^{-12}}$.
2. Express $3(t-1)^{\frac{1}{2}} + 2(t-1)^{\frac{3}{2}}$ in the form $(t-1)^p (qt + r)$.
3. Solve the equation $3(t-1)^{\frac{1}{2}} + 2(t-1)^{\frac{3}{2}} = 0$.
4. Simplify $\sqrt{a^4b^6} \div \sqrt[3]{a^9b^{-3}}$.
5. Write $7(t-3)^{\frac{1}{2}} - 2(t-3)^{\frac{3}{2}}$ in the form $(t-3)^p(kt + m)$.

Tip: When simplifying expressions involving radicals, always look to express them in their simplest forms by factoring out common terms.