Basic Power Rules

Task goals

• Multiply two or more powers of the same base by adding the exponents, writing the result as a single power (e.g. x^2 x x^3, a^4 x a^2, m^5 x m, d^4 x d).
• Divide two powers of the same base by subtracting the exponents (e.g. y^7 / y^2, b^6 / b^5, p^9 / p^3, h^10 / h^6).
• Simplify a power raised to another power by multiplying the exponents (e.g. (k^3)^2, (c^4)^3, (n^2)^5, (e^3)^4).
• Find a missing exponent from a true statement and judge whether a given statement follows the rules (e.g. q^2 x q^5 = q^?, (f^?)^2 = f^10, true or false that s^3 x s^4 = s^7).
• Apply all three rules to expressions that include a numeric coefficient, operating on the coefficient and the exponent separately (e.g. 2a^4 x 3a^2 = 6a^6, 12b^7 / 3b^2 = 4b^5, (3c^3)^2 = 9c^6, (2e^4)^3).
• Recognise that dividing equal powers of the same base gives the zero index, and that a non-zero base to the power zero equals 1 (e.g. p^8 / p^8 = p^0 = 1).
• Combine two or more rules across a multi-step expression, including expressions written as a fraction (e.g. x^2 x x^5 / x^3, (n^5)^2 / n^3, (c^7 x c^2)/c^5, ((n^3)^2 x n^4)/n^5, (24p^8 / 4p^3) / (p^2)^2).
• Reason about the power rules by explaining why a worked step is wrong, locating an error, comparing two expressions, finding a missing factor, and constructing an expression that simplifies to a given power (e.g. why (t^3)^2 = t^5 is wrong, the error in k^4 / k^2 = k^2 / k^2 = 1, 3x^2 x ? = 12x^7, build an expression equal to z^6).