Question

Find two solutions to the equation (x^3 − 64)(x^5 − 1) = 0.

Answer 1

the two roots are x = 1 and x = 4

Explanation:

Data provided in the question:

(x³ − 64) (x⁵ − 1) = 0.

Now,

for the above relation to be true the following condition must be followed:

Either (x³ − 64) = 0 ............(1)

or

(x⁵ − 1) = 0 ..........(2)

Therefore,

considering the first equation, we have

(x³ − 64) = 0

adding 64 both sides, we get

x³ − 64 + 64 = 0 + 64

or

x³ = 64

taking the cube root both the sides, we have

∛x³ = ∛64

or

x = ∛(4 × 4 × 4)

or

x = 4

similarly considering the equation (2) , we have

(x⁵ − 1) = 0

adding the number 1 both the sides, we get

x⁵ − 1 + 1 = 0 + 1

or

x⁵ = 1

taking the fifth root both the sides, we get

`\sqrt[5]{x^5}=\sqrt[5]{1}`

also,

1 can be written as 1⁵

therefore,

`\sqrt[5]{x^5}=\sqrt[5]{1^5}`

or

x = 1

Hence,

the two roots are x = 1 and x = 4