Final answer:
The student's question involves expressing a number as a product of prime factors in index form, specifically 3 x 5² x 2³. This is a high school level mathematics question which involves understanding integer powers, exponents, and the multiplication of exponentiated quantities.
Step-by-step explanation:
The question asks to express the number n as a product of its prime factors in index form, where n is given as 3 x 5² x 2³. When a number is expressed this way, it means that it is the product of prime numbers raised to certain powers, known as exponents. The notation 5², for instance, means 5 multiplied by itself, which is equal to 25. Similarly, 2³ represents 2 multiplied by itself three times, which is 2 x 2 x 2, equalling 8. Therefore, the number n can be rewritten as 3 * 25 * 8.
When raising numbers to powers, we use index notation to simplify expressions. For example, integer powers such as 4³ are a shorthand for writing the number 4 multiplied by itself three times (4 x 4 x 4). Cubing of exponentials involves not only multiplying the base number by itself three times but also multiplying the exponent by 3 if any exponent is present. Furthermore, when dealing with products of numbers with the same base raised to exponents, we can multiply the exponents to find the final power, as seen in Example A.7.1.