1. **Problem:** Find the simplest index form of 81. Step 1: Express 81 as a power of a prime number. Since $81 = 3 \times 3 \times 3 \times 3 = 3^4$. Step 2: Check the options: - a) $g^2$ (not related) - b) $3 \times 2 \times 3 \times 3$ (equals $3 \times 2 \times 3 \times 3 = 54$, not 81) - c) $5^3 = 125$ (not 81) - d) Both b and c (both incorrect) **Answer:** None of the options correctly represent 81 as a simplest index form. The simplest index form is $3^4$. 2. **Problem:** Identify the term unlike the others in each sum. (1) Terms: $2x, 4x, 6x, 36, 8x$ - All terms except 36 contain the variable $x$. - 36 is a constant, so it is unlike the others. (2) Terms: $m, 5mi, Geel, 30mi$ - $m, 5mi, 30mi$ contain variables or variable terms. - "Geel" is a word, not a mathematical term, so it is unlike the others. 3. **Problem:** Given $P(x) = 8x^3 - 3x^4 + 4x^2 - 7$ and $H(x) = x^4 - x^2 + 6$, find: a) $P(x) + H(x)$ Step 1: Add corresponding terms: $$P(x) + H(x) = (8x^3 - 3x^4 + 4x^2 - 7) + (x^4 - x^2 + 6)$$ Step 2: Combine like terms: $$(-3x^4 + x^4) + 8x^3 + (4x^2 - x^2) + (-7 + 6) = -2x^4 + 8x^3 + 3x^2 - 1$$ b) $P(x) - H(x)$ Step 1: Subtract corresponding terms: $$P(x) - H(x) = (8x^3 - 3x^4 + 4x^2 - 7) - (x^4 - x^2 + 6)$$ Step 2: Combine like terms: $$-3x^4 - x^4 + 8x^3 + 4x^2 + x^2 - 7 - 6 = -4x^4 + 8x^3 + 5x^2 - 13$$ 4. **Problem:** Solve the quadratic equation $x = 9x^2 - 64 + 2x^2$. Step 1: Combine like terms on the right: $$x = (9x^2 + 2x^2) - 64 = 11x^2 - 64$$ Step 2: Rearrange to standard form: $$11x^2 - x - 64 = 0$$ Step 3: Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=11$, $b=-1$, $c=-64$: $$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 11 \times (-64)}}{2 \times 11} = \frac{1 \pm \sqrt{1 + 2816}}{22} = \frac{1 \pm \sqrt{2817}}{22}$$ Step 4: Simplify: $$x = \frac{1 \pm \sqrt{2817}}{22}$$ 6. **Problem:** Factorise the denominator $2\sqrt{5} + 2$ in the expression $\frac{5\sqrt{5}}{2\sqrt{5} + 2}$. Step 1: Factor out the common factor 2: $$2\sqrt{5} + 2 = 2(\sqrt{5} + 1)$$ **Answer:** The denominator factorises as $2(\sqrt{5} + 1)$.