If 𝑥 equals one-fifth, 𝑦 equals
one-half, and 𝑧 equals two-fifths, find the value of 𝑥 to the fourth power divided
by 𝑦𝑧 squared.
We’ve been asked to evaluate this
expression, which involves powers, products, and division of three fractions. Let’s begin by substituting the
values of 𝑥, 𝑦, and 𝑧 into the expression. This gives one-fifth to the fourth
power divided by one-half multiplied by two-fifths all squared. We can then evaluate the
product. One-half multiplied by two-fifths
is equal to one-fifth. So the entire expression simplifies
to one-fifth to the fourth power divided by one-fifth squared.
We now observe that we are dividing
two powers of the same base. In this case, the base is the
fraction one-fifth. We can recall the division law for
exponents, which tells us that when we are dividing two powers of a nonzero base, we
subtract the exponents. So 𝑎 to the 𝑚th power divided by
𝑎 to the 𝑛th power is 𝑎 to the 𝑚 minus 𝑛th power. Applying this law with 𝑎 equal to
one-fifth, 𝑚 equal to four, and 𝑛 equal to two gives one-fifth to the power of
four minus two, which simplifies to one-fifth squared.
To simplify further, we need to
evaluate one-fifth squared, which we can do in two ways. One method would simply be to
calculate one-fifth multiplied by one-fifth. Recalling that when we multiply
fractions, we multiply the numerators together and multiply the denominators
together, gives one over 25. Alternatively, we could recall
another law of exponents, which is that when we raise a fraction to a power, this is
equivalent to raising the numerator and denominator separately to that power. So one-fifth squared is the same as
one squared over five squared. And evaluating the squares gives
one over 25 again.
Both methods of course give the
same result, which is that if 𝑥 equals one-fifth, 𝑦 equals one-half, and 𝑧 equals
two-fifths, then the value of 𝑥 to the fourth power divided by 𝑦𝑧 squared is one