Final answer:
The two expressions that are equivalent to z^(8) according to the rules of exponents are (B) (z*z^(3))^(2) and (C) z^(6)*z^(2), because the exponents when multiplied or added respectively result in z^(8).
Step-by-step explanation:
The question asks to select two expressions that are equivalent to z^(8). To solve this, we must use the rules for manipulating exponents such as when multiplying or dividing exponential expressions, and when raising an exponential expression to a power.
Option Analysis
- (A) ((z^(12))/(z^(10)))^(3): Here we first divide the exponents (subtracting the powers) to get z^(2). Then cubing the result (multiplying the power by 3), we get z^(6), which is not equal to z^(8).
- (B) (z*z^(3))^(2): Multiplying the exponents inside the parenthesis gets us z^(1+3), which is z^(4). Squaring that (multiplying the power by 2) results in z^(8), which is correct.
- (C) z^(6)*z^(2): Simply add the powers because the bases are the same, resulting in z^(8), which is correct.
- (D) (z^(16))/(z^(2)): Subtract the exponent of the denominator from the numerator to get z^(14), which is not z^(8).
- (E) ((z^(6))/(z))^(2): Subtracting the exponents, we get z^(5). Squaring that would give us z^(10), which is not z^(8).
Therefore, the two expressions equivalent to z^(8) are options B and C.