Advanced Power Rules

Task goals

• Write and evaluate numbers with a negative index as a fraction or decimal (e.g. 2^-1, 10^-3, 5^-1, 4^-2).
• Write and evaluate numbers with a fractional index as roots (e.g. 16^(1/2), 27^(1/3), 81^(1/2)).
• Extend negative- and fractional-index rules to simple algebraic terms and connect fractional-index notation to root notation (e.g. a^-3 = 1/a^3, x^(1/2) = sqrt(x)), including comparing a negative-index value to 1.
• Evaluate and simplify expressions that combine negative indices with multiplying or dividing powers, and convert between index and root or surd form (e.g. 2^-3, a^4 x a^-2, 32^(1/5), writing x^-2 with a positive index).
• Apply the power-of-a-power rule to simplify expressions with fractional indices (e.g. (m^3)^2, (p^4)^(1/2), q^(2/3) = (cube root of q)^square).
• Simplify more complex algebraic expressions containing multiple negative and fractional indices, giving answers with positive indices (e.g. x^-3 y^2 x x^5, a^7/a^-2, (m^3 n^-1)/(m^-2 n^4)).
• Evaluate negative fractional powers and simplify power-of-a-power expressions with negative or fractional indices to positive-index form (e.g. 16^(-1/2), 27^(-2/3), (p^-2)^3, writing r^(-1/2) with a root).
• Reason about and solve problems involving negative and fractional indices, including comparisons, true/false justification, and solving equations (e.g. compare 2^-3 and 2^-4, decide if x^-2 = -x^2, solve v^(1/2) = 5, find u^-1 given u^3 = 8).