We want to find the real solution
of the equation x cubed equals negative 64.
So to solve for x, we'll need to undo the cubing on x
which means one way to solve for x will be to take
the cubed root of both sides of the equation.
So we'd have the cube root of x cubed equals the cube root
of negative 64.
On the left side the cube root of x cubed is equal to just
one factor of x again because the cube root undoes
the cubing.
And on the right side, we're going to simplify the cubed
root of negative 64.
If we don't recognize negative 64 as a perfect cube what we
can do is look at the prime factorization of positive 64
and then work with the signs
so that the product would be negative 64.
So let's first find the prime factorization of positive 64.
64 is equal to eight times eight because 64 is a perfect
square, eight is equal to four times two, two is prime.
And four is equal to two times two.
So notice that positive 64 is equal to six factors of two.
And because we're looking for a perfect cubed,
notice how if we pair the two, we have three groups
where each group has two factors of two in it
and therefore we know that 64 is equal to four times
four times four which equals four cubed.
So positive 64 is a perfect cube.
And then from here, because we're trying to find
the cubic of negative 64 notice how if we made each
of these factors negative,
we would have negative four times negative four
times negative four, and when multiplying if we have
an odd number of negatives, the product is negative
and therefore negative 64 is a perfect cube,
it's equal to the cube of negative four.
So we can write the right side as the cube root
of the cube of negative four and again the cube root undoes
the cubing and we have our solution x equals negative four.
Let's check our solution in the equation,
substituting negative four for x, we would have the cube
of negative four equals negative 64, which is true.
Verifying our solution is correct.
In the answer cell we only enter negative four.
But let's solve this equation again not using a radical,
but a rational exponent.
So beginning with the original equation again,
we have x cubed equals negative 64, remember the power
to power property of exponents tells us that if we have A
raised to the power of M, raised to the power of N,
this is equal to A raised to the power of M times N.
So another way to undo the cubing here would be to raise
both sides of the equation to the reciprocal power of three
or to the one third power.
So if you raise both sides of the equation to the one third
power, we can also solve for x.
Notice on the left side, we have powers to powers
so I multiply the exponents three times one third is one
and x to the first is just x.
On the right side, we have negative 64 raised
to the one third power, so if I wanted to we could now
write the rational exponent as a radical
where the denominator tells us the index so we have
the cube root of negative 64 and the exponent
on the negative 64 would be one so we can leave that off.
And just as before we have x equals the cubic
of negative 64, which we already know is equal
to negative four.
So we can solve this equation using radicals
or rational exponents.
I hope you found this helpful.