Final answer:
The expression for 5n², given n = 3 x 5² x³, is expressed as a product of its prime factors in terms of x as 5⁵ x 3² x x⁶.The expression for 5n² is 5⁵ * 3² * x⁶, expressed as a product of prime factors in terms of x.
Step-by-step explanation:
Given the expression for n, which is n = 3 x 5² x³, we want to express 5n² as a product of its prime factors in terms of x. To find 5n², we first square n, which involves squaring each component in the expression for n.
Squaring 3 gives us 3², squaring 5² gives us 5⁴, and squaring x³ gives us x⁶. We then multiply these together with the 5 that is outside the square in the expression 5n². Therefore, we have:
5 x (3²) x (5⁴) x (x⁶) = 5 x 9 x 625 x x⁶.
Combining the terms with the same base, we get:
5 x 5⁴ x 9 x x⁶ = 5⁵ x 9 x x⁶.
Now, noting that 9 is 3², which is also a prime factor, we can finally write:
5n² = 5⁵ x 3² x x⁶.
This expression is now fully expressed as a product of prime factors and in terms of x.