High School

Match the equivalent expressions.

1. [tex]$x^2$[/tex]
2. [tex]$x^4$[/tex]
3. [tex]$x^6$[/tex]
4. [tex]$x^8$[/tex]

A. [tex]$2x^6$[/tex]
B. [tex]$4x^6$[/tex]
C. [tex]$2x^9$[/tex]
D. [tex]$4x^5$[/tex]

Answer :

Certainly! Let's match the equivalent expressions step by step.

We need to match three expressions given in terms of powers of [tex]\( x \)[/tex]:

1. [tex]\( x^2 \)[/tex]
2. [tex]\( x^6 \)[/tex]
3. [tex]\( x^8 \)[/tex]

with the following options:

- [tex]\( 2x^6 \)[/tex]
- [tex]\( 4x^6 \)[/tex]
- [tex]\( 2x^9 \)[/tex]
- [tex]\( 4x^5 \)[/tex]

Let's look at each one to see which matches:

1. [tex]\( x^2 \)[/tex]:
- None of the options are suitable to match [tex]\( x^2 \)[/tex] as most involve higher powers of [tex]\( x \)[/tex] combined with constants.

2. [tex]\( x^6 \)[/tex]:
- [tex]\( x^6 \)[/tex] could logically pair with the given option that modifies [tex]\( x^6 \)[/tex] which is [tex]\( 2x^6 \)[/tex].
- This is because [tex]\( x^6 \)[/tex] multiplied by 2 maintains the same power: [tex]\( x^6 \cdot 2 = 2x^6 \)[/tex].

3. [tex]\( x^8 \)[/tex]:
- [tex]\( x^8 \)[/tex] can be written as [tex]\( x^5 \cdot x^3 = 4x^5 \cdot x^3/4 \)[/tex]. Rechecking the matching expressions:
- Thus, [tex]\( x^8 \)[/tex] simplifies to an equation directly involving a combination with [tex]\( x^5 \)[/tex].
- On simplifying, [tex]\( x^8 \)[/tex] is effectively represented by the combination that leads to a scaled version of [tex]\( x^5 \)[/tex], hence the equivalent expression is [tex]\( 4x^5 \)[/tex].

Thus, the equivalent expressions are:

- [tex]\( x^2 \)[/tex] doesn't have an appropriate match among these given options.
- [tex]\( x^6 \)[/tex] matches with [tex]\( 2x^6 \)[/tex].
- [tex]\( x^8 \)[/tex] matches with [tex]\( 4x^5 \)[/tex].

Therefore, the equivalences we match will be:
- [tex]\( x^6 \)[/tex] with [tex]\( 2x^6 \)[/tex]
- [tex]\( x^8 \)[/tex] with [tex]\( 4x^5 \)[/tex]

These are the matched equivalences based on our analysis.