Explanation:
remember, when we multiply terms with the same base and exponents, the result is the same base with the exponent being the sum of the individual exponents.
and when we divide, we subtract the exponents.
why is that ?
keep in mind, an exponent just means how many time the base value is multiplied by itself.
e.g.
a⁵ = a×a×a×a×a
and therefore
e.g.
a⁶×a³ = a×a×a×a×a×a × a×a×a = a⁹
it really is that simple.
m⁹n² / (m^-4 × n⁷)
can be seen as
m⁹/(m^-4) × n²/n⁷
we are only dealing with multiplication. multiplication is associative and commutative.
a × b = b × a
(a × b) × c = a × (b × c)
and then
m⁹/(m^-4) = m^(9-4) = m⁵
n²/n⁷ = n^-5 = 1/n⁵
so, the simplified expression is
m⁵ × 1/n⁵ = m⁵/n⁵
which is the same as
(m/n)⁵
waste per person :
waste produced
9.665 × 10¹¹ pounds
by
4.38 × 10⁸ pounds
so, the pounds of waste per person is
9.665 × 10¹¹ / (4.38 × 10⁸) = 9.665/4.38 × 10¹¹/10⁸ =
= 2.206621005... × 10^(11-8) =
= 2.206621005... × 10³ ≈ 2.21 × 10³
(w³/6)^-3
for the same reason of what an exponent stands for, for an exponent of an exponent we multiply the individual exponents.
e.g.
(a²)³ = a×a × a×a × a×a = a⁶
again, it is all the same simple basic concept.
(w³/6)^-3 = (w^(3×-3)) / (6^(1×-3)) =
= (w^-9) / (6^-3)
remember, a negative exponent just means 1/...
so, our result is
1/w⁹ / 1/6³ = 6³/w⁹
which is the same as
(6/w³)³
exponential function yes/ no :
If your function can be written in the form
y = ab^x, where a and b are constants,
a <> 0, b > 0 and b <> 1, then it must be exponential.
under the assumption it is an exponential function the point x = 0, y = 6 tells us
6 = ab⁰
because b⁰ = 1, this means a = 6.
the next point x = 1, y = 9 would then tell us
9 = 6×b¹
b = 9/6 = 3/2
now we need to check, if the following points follow this :
x = 2, y = 27
27 = 6×(3/2)² = 6×9/4 = 54/4 = 13.5
that is wrong. 27 is NOT 13.5
so, it is not an exponential function.
foxes :
y = 50 × 3^x
with x being the number of 13-year periods.
26 years are two 13-year periods.
so, we have
y = 50 × 3² = 50×9 = 450
after 26 years there will be 450 foxes.