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Start with the given equation.
Rewrite 64 as
.
From here you can see that the answer is 2, but let's keep going
Take the 6th root of both sides. This will eliminate the exponent "6"
Take the 6th root of
to get "z". Take the 6th root of
to get 2
So the solution is
======================================
Here's another way to solve for "z"
Start with the given equation.
Subtract 64 from both sides.
Rewrite 64 as
.
. Rewrite
as
. Rewrite
as
Now let
and
So we now have
Factor using the difference of squares formula
Plug in
and
Factor the first binomial
using the sum of cubes formula
Factor the second binomial
using the sum of cubes formula
Rearrange the terms
Now set each factor equal to zero:
,
,
or
So our first two solutions are
or
Now let's solve
Start with the given equation.
Notice we have a quadratic equation in the form of
where
,
, and
Let's use the quadratic formula to solve for z
Start with the quadratic formula
Plug in
,
, and
Negate
to get
.
Square
to get
.
Multiply
to get
Subtract
from
to get
Multiply
and
to get
.
Simplify the square root (note: If you need help with simplifying square roots, check out this
solver)
Break up the fraction.
Reduce.
or
Break up the expression.
So the next two solutions are are
or
which approximate to
or
-----------------------------------------------
Now let's solve
Start with the given equation.
Notice we have a quadratic equation in the form of
where
,
, and
Let's use the quadratic formula to solve for z
Start with the quadratic formula
Plug in
,
, and
Square
to get
.
Multiply
to get
Subtract
from
to get
Multiply
and
to get
.
Simplify the square root (note: If you need help with simplifying square roots, check out this
solver)
Break up the fraction.
Reduce.
or
Break up the expression.
So the next two solutions are
or
which approximate to
or
========================================================
Answer:
So altogether, we have the 6 solutions
,
,
,
,
or
(