Express the following in index form: (2)/(3) x (2)/(3) x (2)/(3)

$$5^X = 5^2$$

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. $$125^{1/3} = \sqrt[3]{125} = \boxed{5}$$