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1. Integers

9. Here is an equation: [tex]x^3 - x = 120[/tex]

a. Is [tex]x = 5[/tex] a solution? Give a reason for your answer.

b. Is [tex]x = -5[/tex] a solution? Give a reason for your answer.

10.

a. Write 64 as a product of its prime factors.

b. Show that 64 is a square number and a cube number.

c. Write 729 as a product of prime factors.

d. Show that 729 is both a square number and a cube number.

e. Find another integer that is both a square number and a cube number.

11. Look at the following solution of the equation [tex]x^6 = 64[/tex]

[tex]x^6 = 64[/tex]

Sagot :

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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.
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