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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.
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.
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