SOLUTION: I need to find z when z^6=64. I know the answer is 2 but what are the steps to solve for z?

Question 151647This question is from textbook PreAlgebra
: I need to find z when z^6=64. I know the answer is 2 but what are the steps to solve for z? This question is from textbook PreAlgebra

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
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
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