Q1. The answer is 1 over 5 to the 3rd power.
5 to the 5th is 5⁵
5 to the 8th is 5⁸
5 to the 5th over 5 to the 8th is [tex] \frac{ 5^{5} }{ 5^{8} } [/tex]
To simplify, we will use two rules:
[tex] \frac{ x^{a}}{ x^{b} } = x^{a-b} \\ x^{-a} = \frac{1}{ x^{a}} [/tex]
Therefore:
[tex]\frac{ 5^{5} }{ 5^{8} } = 5^{5-8} =5^{-3}= \frac{1}{5^{3}} [/tex]
[tex] \frac{1}{5^{3}} [/tex] is the same as 1 over 5 to the 3rd power.
Q2. The answer is (53)−4.
The exponents are multiplied when are expressed in the form:
[tex] (x^{a}) ^{b}= x^{a*b}[/tex]
So, [tex] (5^{3} )^{-4}=5^{3*(-4)}=5^{-12}[/tex]
Other choices are incorrect:
one third to the 4th times one third to the 7th is [tex]( \frac{1}{3} )^{4} *( \frac{1}{3}
)^{7}=( \frac{1}{3} )^{4+7}[/tex]
3 to the 7th over 3 to the 15th is [tex] \frac{ 3^{7} }{3^{15}} =3^{7-15}[/tex]
45 ⋅ 42 is [tex] 4^{5}* 4^{2} = 4^{5+2} [/tex]
