Calculate negative three and
one-fifth to the seventh power multiplied by negative one and a half to the sixth
power all over negative 16 over five to the sixth power multiplied by negative three
over two to the fourth power, giving your answer in its simplest form.
We could evaluate this expression
by evaluating each power in turn and then multiplying the fractions. However, it will be simpler to use
some laws of exponents to simplify the expression before evaluating any powers.
Let’s begin by rewriting each mixed
number as a fraction. Negative three and one-fifth is
equal to negative 16 over five. And negative one and a half is
equal to negative three over two. We now see that we actually have
common bases in the numerator and denominator of this quotient. As we are dividing powers of the
same rational bases, we can recall the division law of exponents. This states that when we divide
powers of the same rational base, we subtract the powers. 𝑎 to the 𝑚th power divided by 𝑎
to the 𝑛th power is 𝑎 to the power of 𝑚 minus 𝑛.
Applying this law to each pair of
common bases gives negative 16 over five to the power of seven minus six multiplied
by negative three over two to the power of six minus four. Simplifying the exponents gives
negative 16 over five to the first power multiplied by negative three over two
Now, as the base in the first term
is raised to the first power, the whole term is just equal to the base itself. For the second term, we can recall
that when we raise a fraction with a nonzero denominator to a power, this is
equivalent to raising the numerator and denominator separately to that power. So, the second term in the product
can be written as negative three squared over two squared. Evaluating the squares gives nine
Before we multiply the two
fractions, we can simplify by cross canceling a factor of four to give negative four
over five multiplied by nine over one. Finally, multiplying using the
usual rules of fractions gives negative 36 over five. So, by first using the laws of
exponents to simplify the given expression, we’ve found that its value is negative
36 over five.