Video Transcript
If 𝑥 is equal to the cube root of
375 and 𝑦 is equal to the cube root of 81, find the value of 𝑥 plus 𝑦 all
cubed.
In this question, we are given the
values of 𝑥 and 𝑦 as radicals and are asked to evaluate an expression involving 𝑥
and 𝑦. We can begin by substituting the
given values into the expression to obtain the cube root of 375 plus the cube root
of 81 all cubed. At this point, there are two
options for evaluating the expression. Either we can attempt to simplify
the expression inside the parentheses or we can expand the cube of the terms in the
parentheses.
Of course, expanding the cube will
make the expression quite complicated. So, we will start by trying to
simplify the expression inside the parentheses. To do this, we need to simplify
each cube root. We can do this by recalling that we
can take the cube roots of any factor separately. In other words, for any real values
𝑎 and 𝑏, the cube root of 𝑎 times 𝑏 is equal to the cube root of 𝑎 times the
cube root of 𝑏.
We can use this property to reduce
the size of the numbers inside the cube roots. For instance, we can see that 375
is equal to five cubed times three. Similarly, we can calculate that 81
is three cubed times three. This allows us to rewrite the
expression we want to evaluate as the cube root of five cubed times three plus the
cube root of three cubed times three all cubed. This is useful since we can take
the cube root of each factor separately. And we know that the cube root of
𝑎 cubed will be equal to 𝑎. This means we can take cubic
factors out of the radical by taking their cube roots.
First, we can set 𝑎 equal to five
cubed and 𝑏 equal to three to get that the cube root of five cubed times three is
equal to the cube root of five cubed times the cube root of three. We can follow the same process for
the second term inside the parentheses. We set 𝑎 equal to three cubed and
𝑏 equal to three to get the cube root of three cubed times the cube root of
three. We then need to cube the sum of
these terms.
We can now simplify the expression
inside the parentheses by evaluating the cube roots. We have that the cube root of five
cubed is five and the cube root of three cubed is three. So, we obtain five times the cube
root of three plus three times the cube root of three all cubed. We can now see that the two terms
inside the parentheses share a factor of the cube root of three. So, we can take out this factor and
add them together by adding five and three. This gives us eight times the cube
root of three all cubed.
We now have a product raised to an
exponent. So, we can now evaluate this
expression using the laws of exponents. In particular, we know that for any
real numbers 𝑎 and 𝑏 and integer 𝑛, 𝑎 times 𝑏 all raised to the power of 𝑛 is
equal to 𝑎 raised to the power of 𝑛 times 𝑏 raised to the power of 𝑛 provided
that these are all well defined. We can apply this result with 𝑎
equal to eight, 𝑏 equal to the cube root of three, and 𝑛 equal to three to get
eight cubed times the cube root of three cubed.
Finally, we can evaluate. We calculate that eight cubed is
512 and the cube root of three cubed is three, giving us 512 times three, which we
can calculate is 1,536.
