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

Note that:

a.

$10=5\cdot2$

Recall the law of exponents for a multiplication operating inside of parentheses:

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

According to this, we obtain the following:

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

We want to use the knowledge in 'a' in order to obtain terms with identical bases in both the numerator and denominator,

Let's return to the problem and apply the knowledge obtained from both 'a' and 'b':

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

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 proceeded to simplify the first fraction in the multiplication.

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

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

Apply this law to the expression as shown below:

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

In the first stage we applied this law to the fraction in the multiplication and then proceeded to simplify the expression that we obtained,

Let's summarize the various steps of the solution:

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

Therefore, the correct answer is answer b.