Illustrate a meticulously detailed mathematical symbol for cube root. It has an empty space on the top that signifies the unknown value of n. Within the cube root symbol, neatly inscribed, are the numbers 1.25 and 10.

The product 1.25 X 10^n is the cube of an integer when n=

A. 1999
B. 2000
C. 2001
D. 2002

To determine the value of n when the product 1.25 × 10^n is the cube of an integer, we need to analyze the given expression and understand the properties of cubes.

Let's start by simplifying the expression.

The cube of an integer can be represented as the product of that integer three times. So, if 1.25 × 10^n is the cube of an integer, we can write it as:

(1.25 × 10^n) = (x^3)

To solve for n, we need to eliminate the decimal and the scientific notation. We can do that by representing 1.25 as a fraction:

(1.25 × 10^n) = (5/4 × 10^n) = (x^3)

Now, to get rid of the fraction, we can multiply both sides of the equation by 4:

(4 × 1.25 × 10^n) = (4 × x^3)
5 × 10^n = (2x)^3 #(Multiplying 1.25 by 4 = 5 and x by 2 = 2x)

Now, we can write 10^n as (10^3)^m, where m represents a new integer:

5 × (10^3)^m = (2x)^3

To equate the powers of 10, we can compare the exponents:

5 × 10^(3m) = (2x)^3

From here, we can see that the left-hand side is divisible by 5, but the right-hand side contains a cube. For the equation to hold true, the left-hand side must also be divisible by 5.

This tells us that the exponent of 10, 3m, must be divisible by 5.

Looking at the answer choices, we see that the only option where 3m is divisible by 5 is when n = 2000.

Therefore, the answer is B. 2000.

since 5^3 = 125 = 1.25x10^2

any power of 10 will do, as long as it is 2 more than a multiple of 3.

That's because each 3 added to the power is the same as multiplying by 1000, which is 10^3.