Video Transcript
If 𝑥 is equal to three over two
and 𝑦 is equal to negative four over five, find the value of 𝑥 squared 𝑦 minus
𝑥𝑦 cubed, giving your answer as a fraction in simplest form.
We will begin this question by
substituting the values of 𝑥 and 𝑦 into our expression. 𝑥 squared 𝑦 minus 𝑥𝑦 cubed is
therefore equal to three over two squared multiplied by negative four over five
minus three over two multiplied by negative four over five cubed. Our next step is to evaluate the
powers by recalling 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 over 𝑏
to the 𝑛th power. This means that three over two
squared is equal to three squared over two squared, which is equal to nine over
four. In the same way, negative four over
five cubed is equal to negative four cubed over five cubed. And this is equal to negative 64
over 125. Our expression therefore simplifies
to nine over four multiplied by negative four over five minus three over two
multiplied by negative 64 over 125.
We can simplify the first part of
our expression by dividing the numerator and denominator by four. In the same way, we can divide the
numerator and denominator of the second part of our expression by two. And our expression simplifies to
negative nine-fifths minus negative 96 over 125. This in turn simplifies to negative
nine over five plus 96 over 125. Finally, in order to add fractions,
we must have the same denominators, so we multiply the numerator and denominator of
our first fraction by 25. The expression becomes negative 225
over 125 plus 96 over 125. Finally, we add the numerators,
giving us negative 129 over 125. Since there are no shared factors
apart from one, this is the value of 𝑥 squared 𝑦 minus 𝑥𝑦 cubed as a fraction in
its simplest form.
