Let's break down and solve the problems step by step.
### 9. [tex]\( x^3 - x = 120 \)[/tex]
a. Is [tex]\( x = 5 \)[/tex] a solution? Give a reason for your answer.
Substitute [tex]\( x = 5 \)[/tex] into the equation:
[tex]\[ 5^3 - 5 = 125 - 5 = 120 \][/tex]
So, [tex]\( x = 5 \)[/tex] satisfies the equation [tex]\( x^3 - x = 120 \)[/tex]. Therefore, [tex]\( x = 5 \)[/tex] is a solution.
b. Is [tex]\( x = -5 \)[/tex] a solution? Give a reason for your answer.
Substitute [tex]\( x = -5 \)[/tex] into the equation:
[tex]\[ (-5)^3 - (-5) = -125 + 5 = -120 \][/tex]
So, [tex]\( x = -5 \)[/tex] does not satisfy the equation [tex]\( x^3 - x = 120 \)[/tex]. Therefore, [tex]\( x = -5 \)[/tex] is not a solution.
### 10.
a. Write 64 as a product of its prime factors.
64 can be written as a product of its prime factors:
[tex]\[ 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 \][/tex]
b. Show that 64 is a square number and a cube number.
A square number is a number that can be written as [tex]\( n^2 \)[/tex] for some integer [tex]\( n \)[/tex]. A cube number is a number that can be written as [tex]\( n^3 \)[/tex].
- Square number: Since [tex]\( 64 = 8^2 \)[/tex], 64 is a square number.
- Cube number: Since [tex]\( 64 = 4^3 \)[/tex], 64 is a cube number.
Therefore, 64 is both a square number and a cube number.
c. Write 729 as a product of prime numbers.
729 can be written as a product of its prime factors:
[tex]\[ 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \][/tex]
d. Show that 729 is both a square number and a cube number.
- Square number: Since [tex]\( 729 = 27^2 \)[/tex], 729 is a square number.
- Cube number: Since [tex]\( 729 = 9^3 \)[/tex], 729 is a cube number.
Therefore, 729 is both a square number and a cube number.
e. Find another integer that is both a square number and a cube number.
An integer that is both a square number and a cube number must be a number that is a power of 6, because it must be both [tex]\( n^2 \)[/tex] and [tex]\( m^3 \)[/tex], which means it needs to be [tex]\( k^6 \)[/tex] for some integer [tex]\( k \)[/tex].
After 64, the next number of this type is:
[tex]\[ 2^3 \times 3^3 = (2 \times 3)^3 = 6^3 = 216 \][/tex]
So, 216 is another integer that is both a square number and a cube number.
