Question 13

27^x can be expressed as (p/3)^3.

To express 27^x in terms of p, we can rewrite 27 as 3^3, so 27^x becomes (3^3)^x, which simplifies to 3^(3x). Given that p = 3^(x+1), we can derive p/3 and subsequently raise it to the power of 3 to find the relationship.

A) sqrt(3)p

Option A is incorrect because, although it includes p, it does not correctly relate to 27^x. The expression sqrt(3)p does not align with the transformation we need to make with the base of 3 or the exponent of x.

B) 3p

Option B is also incorrect. While it incorporates p, it does not accurately represent 27^x, which requires a cubic relationship due to the exponent of 3 in 27. The term 3p does not capture the necessary exponentiation of x.

C) (3p)^3

Option C is incorrect as it suggests an incorrect transformation. While it does use p, it multiplies it by 3 and raises the entire expression to the third power, which does not correspond to the expression for 27^x derived from the given p.

D) (p/3)^3

Option D is correct because it accurately rephrases 27^x in terms of p. By substituting p = 3^(x+1), dividing by 3 gives us 3^x, and cubing this results in 27^x, aligning perfectly with our transformation.

E) (p/9)^3

Option E is incorrect as it misrepresents the relationship between p and 27^x. Dividing p by 9 introduces an unnecessary factor, which does not reflect the exponentiation necessary to arrive at 27^x.

Conclusion

The expression (p/3)^3 correctly captures the relationship between p and 27^x, demonstrating the necessary transformation from the original equation. All other options fail to maintain the correct mathematical relationship needed to express 27^x in terms of p, either by misapplying the operations or failing to align with the base's exponent.