Video Transcript
Given that 𝑥 cubed over eight plus
seven is equal to one, find the value of 𝑥 and state whether it is a rational or
irrational number.
Our first step in solving this
equation is to subtract seven from both sides. As one minus seven is equal to
negative six, we’re left with 𝑥 cubed over eight is equal to negative six. We might be tempted at this point
to multiply both sides of the equation by eight. However, if we notice that eight is
a cube number, it will be easier to cube root both sides. When cube rooting any fraction, we
can cube root the numerator and denominator separately.
The cube root of 𝑎 over 𝑏 is
equal to the cube root of 𝑎 over the cube root of 𝑏. The cube root of 𝑥 cubed is equal
to 𝑥. As two cubed is equal to eight, the
cube root of eight is two. Our equation becomes 𝑥 over two is
equal to the cube root of negative six. Multiplying both sides of this
equation by two gives us a value of 𝑥 equal to two multiplied by the cube root of
negative six. This is the answer to the first
part of the question.
We’re also asked to state whether
this number is rational or irrational. A rational number is any number
that can be written as a fraction where the numerator and denominator are
integers. Conversely, an irrational number
cannot be written as a fraction. Some common irrational numbers are
𝜋, the square of two, and the square root of three. In fact, many roots, square roots,
cube roots, and so on are also irrational numbers. The cube root of six is one such
example. It is equal to 1.817120 and so
on. As its decimal value does not
terminate and is not recurring, the answer will be irrational.
The cube root of any negative
number is always negative. In fact, the cube root of negative
six is equal to the negative value of the cube root of six, negative 1.817120 and so
on. This means that the cube root of
negative six part of our answer is irrational. We are multiplying this by two,
which is a rational number. Multiplying an irrational number by
any rational number always gives us an irrational answer. This means that our value for 𝑥,
two multiplies by the cube root of negative six, is irrational.
