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N = 3 x 5² x³ write 5n² as a product of its prime factors, give your answer in terms of x

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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.

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