Here are the solutions to the questions. Express the following in index form: $\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}$ When a number is multiplied by itself multiple times, it can be written in index form. $$ \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \left(\frac{2}{3}\right)^3 $$ The answer is $\boxed{\left(\frac{2}{3}\right)^3}$. 10. Find the value of $X$: $5^X = 25$ Step 1: Express 25 as a power of 5. $$ 25 = 5^2 $$ Step 2: Equate the powers. $$ 5^X = 5^2 $$ Since the bases are the same, the exponents must be equal. $$ X = \boxed{2} $$ 11. Simplify: i) $5^3 \times 5^4$* Using the rule $a^m \times a^n = a^{m+n}$: $$ 5^3 \times 5^4 = 5^{3+4} = \boxed{5^7} $$ ii) $y^4 \times y^4$* Using the rule $a^m \times a^n = a^{m+n}$: $$ y^4 \times y^4 = y^{4+4} = \boxed{y^8} $$ 12. Simplify and give your answer in index form: i) $(4^2)^3$* Using the rule $(a^m)^n = a^{m \times n}$: $$ (4^2)^3 = 4^{2 \times 3} = \boxed{4^6} $$ ii) $(y^4)^2$* Using the rule $(a^m)^n = a^{m \times n}$: $$ (y^4)^2 = y^{4 \times 2} = \boxed{y^8} $$ 13. Determine the value of: i) $3^0$* Any non-zero number raised to the power of 0 is 1. $$ 3^0 = \boxed{1} $$ ii) $4^0 + 8^0$* Using the rule $a^0 = 1$: $$ 4^0 + 8^0 = 1 + 1 = \boxed{2} $$ iii) $2^0 + 3^0$* Using the rule $a^0 = 1$: $$ 2^0 + 3^0 = 1 + 1 = \boxed{2} $$ 14. Simplify: i) $64^{1/2}$* The power of $\frac{1}{2}$ means taking the square root. $$ 64^{1/2} = \sqrt{64} = \boxed{8} $$ ii) $125^{1/3}$* The power of $\frac{1}{3}$ means taking the cube root. $$ 125^{1/3} = \sqrt[3]{125} = \boxed{5} $$ That's 2 down. 3 left today — send the next one.