Question

Mario simplified the expression (4 z Superscript 8 Baseline) Superscript negative 3 as shown.

(4 z Superscript 8 Baseline) Superscript negative 3 = 4 z Superscript 8 times (negative 3) Baseline = 4 z Superscript negative 24 Baseline = StartFraction 4 Over z Superscript 24 Baseline EndFraction


Which statement explains Mario’s error?
He should have added the exponents instead of multiplying them.
He should have subtracted the exponents instead of multiplying them.
He should have applied the exponent –3 to 4, and not to z, to get 4 Superscript negative 3 Baseline z Superscript 8 Baseline = 4 Superscript negative 3 Baseline z Superscript 8 Baseline = StartFraction z Superscript 8 Baseline Over 64 EndFraction.
He should also have applied the exponent –3 to 4 to get 4 Superscript negative 3 Baseline z Superscript 8 Baseline times (negative 3) Baseline = 4 Superscript negative 4 Baseline z Superscript negative 24 Baseline = StartFraction 1 Over 64 z Superscript 24 EndFraction.

Answer 1

Mario forgot to apply the exponent to the 4. By (ab)^n = a^n b^n,

(4 z^8)^{-3} = 4^{-3} z^{8·(-3)} = 4^{-3} z^{-24} = 1/(64 z^{24}).

So the correct choice is the one saying he should also have applied the exponent −3 to 4 (giving 4^{-3} z^{-24} = 1/(64 z^{24})).