### Video Transcript

What is four over 11 multiplied by
four over 11 multiplied by four over 11 multiplied by four over 11 multiplied by
four over 11 multiplied by four over 11 multiplied by four over 11? Is it option (A) four elevenths to
the power of negative seven? (B) Four elevenths to the power of
seven. (C) Four elevenths to the power of
nine. (D) Seven elevenths to the power of
four. Or (E) twenty-eight elevenths to
the power of seven.

We could evaluate this expression
by multiplying all of the numerators and all of the denominators. This would give us the following
expression which we could calculate with or without a calculator. However, the five options in this
question are given as powers. This means that instead of
evaluating the expression, we can simplify by recalling that repeated multiplication
can be written in exponential form. In particular, in this question, we
are multiplying seven lots of four elevenths. We know that the product of seven
lots of four elevenths can be written by raising four elevenths to an exponent of
seven. And as such, we can conclude that
the correct answer is option (B). The expression in the question is
equivalent to four elevenths to the power of seven.

We will now use the information we
have seen so far to define a key property of the powers of rational numbers. Since a positive integer power of a
rational base is defined by repeated multiplication, we can show that if 𝑛 is a
positive integer and 𝑎 over 𝑏 is a rational number, then 𝑎 over 𝑏 to the 𝑛th
power is equal to 𝑎 to the 𝑛th power divided by 𝑏 to the 𝑛th power. In other words, we can raise the
numerator and denominator to the power separately.