How would one calculate the disk mass of a galaxy with the luminosity of 2.45x`10^(9)` .

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Tamara K. H. | Middle School Teacher | (Level 3) Educator Emeritus

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It has been found that there is a direct relationship between the mass of a star and the luminosity of a star. Luminosity is how much energy a star generates. Hence, if we know the luminosity of one star in a galaxy, we can use that luminosity to approximate the mass of the whole galaxy; however, it should also be known that doing so poses problems and can only give us an approximate mass. The problem stems from the fact that the light emitted can sometimes be only a small fraction of a galaxy's mass; in that case, a galaxy's mass is made up of dark matter as well ("Mass of the Milky Way Galaxy"). However, if we were to use the luminosity of one star to estimate the mass of the entire galaxy, we would use the following equation, where M = Mass, and L = Luminosity:

` ` `L= M^3.5`

To solve for M, meaning mass, we first want to get M standing alone, not L. To get L alone, we first want to know that 3.5 can also be written as the fraction 7/2, giving us L = `M^(7/2)` . To get L alone, we can also use the following law ("Laws of Exponents"):

`X^(m/n) = root(n)(x^(m))`

Using the following law yields us the new equation:
M = `root(7)(L^(2))`

However, to make the above calculation much simpler, we can also understand that radicals can be written as powers of fractions (Purplemath, "Fractional (Rational) Exponents"). For example,  √2 can be written as `2^(1/2)` , and something like `root(10)(25)^(5)` can be written as `(25^(1/10))^(5)` .  Therefore, we can rewrite M = `root(7)(L)^(2)` as M = `(L1/7)2` .

So, to solve for M, we simply plug in our known luminosity and solve the equation:
M = `[(2.45x10^(9))^(1/7)]^(2)`

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