Lonelycloud991idn
Answered

Get the information you need quickly and easily with IDNLearn.com. Our platform is designed to provide accurate and comprehensive answers to any questions you may have.

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 :

GinnyAnsweridn
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.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.