Let's break down each expression step by step as follows:
1. The difference of [tex]\( x \)[/tex] and 4, cubed, times 6:
- This phrase translates to: [tex]\( 6 \cdot (x - 4)^3 \)[/tex]
- Thus, the expression is: [tex]\( 6(x - 4)^3 \)[/tex]
2. 4 less than the 3rd power of the product of 6 and [tex]\( x \)[/tex]:
- First, find the product of 6 and [tex]\( x \)[/tex]: [tex]\( 6x \)[/tex]
- Then, find the 3rd power of this product: [tex]\( (6x)^3 \)[/tex]
- Finally, subtract 4 from this result: [tex]\( (6x)^3 - 4 \)[/tex]
- Thus, the expression is: [tex]\( (6x)^3 - 4 \)[/tex]
3. The product of [tex]\( x \)[/tex] cubed and 6, minus 4:
- First, find [tex]\( x \)[/tex] cubed: [tex]\( x^3 \)[/tex]
- Then, multiply this by 6: [tex]\( 6x^3 \)[/tex]
- Finally, subtract 4 from this product: [tex]\( 6x^3 - 4 \)[/tex]
- Thus, the expression is: [tex]\( 6x^3 - 4 \)[/tex]
4. 4 less than [tex]\( 6x \)[/tex], cubed:
- First, subtract 4 from [tex]\( 6x \)[/tex]: [tex]\( 6x - 4 \)[/tex]
- Then, cube this result: [tex]\( (6x - 4)^3 \)[/tex]
- Thus, the expression is: [tex]\( (6x - 4)^3 \)[/tex]
Now, we will pair each detailed expression with the initial pairs provided in the question:
1. [tex]$(6x - 4)^3$[/tex]:
- Matches with: 4 less than [tex]\( 6x \)[/tex], cubed.
- So, [tex]\( (6x - 4)^3 \longrightarrow (6x - 4)^3 \)[/tex]
2. [tex]$(6x)^3 - 4$[/tex]:
- Matches with: 4 less than the 3rd power of the product of 6 and [tex]\( x \)[/tex].
- So, [tex]\( (6x)^3 - 4 \longrightarrow (6x)^3 - 4 \)[/tex]
3. [tex]$6x^3 - 4$[/tex]:
- Matches with: The product of [tex]\( x \)[/tex] cubed and 6, minus 4.
- So, [tex]\( 6x^3 - 4 \longrightarrow 6x^3 - 4 \)[/tex]
4. [tex]$6(x - 4)^3$[/tex]:
- Matches with: The difference of [tex]\( x \)[/tex] and 4, cubed, times 6.
- So, [tex]\( 6(x - 4)^3 \longrightarrow 6(x - 4)^3 \)[/tex]
Hence, the correct pairs are:
[tex]\[ \left( (6x - 4)^3 \longrightarrow (6x - 4)^3, \right) \][/tex]
[tex]\[ \left( (6x)^3 - 4 \longrightarrow (6x)^3 - 4, \right) \][/tex]
[tex]\[ \left( 6x^3 - 4 \longrightarrow 6x^3 - 4, \right) \][/tex]
[tex]\[ \left( 6(x - 4)^3 \longrightarrow 6(x - 4)^3 \right) \][/tex]
