Solve: Fraction Division with Negative 5 Cubed in Denominator (10/(-5)³)

First, let's note that:

a.

$10=5\cdot2$

For this, we'll recall the law of exponents for multiplication in parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

According to this, we get that:

$(-5)^3=(-1\cdot5)^3=(-1)^3\cdot5^3=-1\cdot5^3=-5^3$

We want to use the understanding in 'a' to get terms with identical bases in the numerator and denominator,

Let's return to the problem and apply the understandings from 'a' and 'b':

$\frac{10}{(-5)^3}=\frac{2\cdot5}{-5^3}=\frac{2}{-1}\cdot\frac{5}{5^3}=-2\cdot\frac{5}{5^3}$

Where in the first stage we used 'a' in the numerator and 'b' in the fraction's denominator, in the next stage we presented the fraction as a multiplication of fractions according to the rule for multiplying fractions, then we simplified the first fraction in the multiplication.

Now we'll use the law of exponents for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

Let's apply this law to the expression we got:

$-2\cdot\frac{5}{5^3}=-2\cdot5^{1-3}=-2\cdot5^{-2}$

where in the first stage we applied this law to the fraction in the multiplication and then simplified the expression we got,

Let's summarize the solution steps:

$\frac{10}{(-5)^3} =-2\cdot\frac{5}{5^3} =-2\cdot5^{-2}$

Therefore, the correct answer is answer b.