Answer :
The value of x is 20.1036 given that x³=8125. This can be obtained by using prime factorization.
Calculate the value of x:
The number 8125 can be written as factors of primes as,
8125 = 5×5×5×5×13
x³ = 5×5×5×5×13
Take cube roots on both sides,
⇒ x = ∛5×5×5×5×13
x = 5∛5×13
x = 5 × 4.02072 ⇒ x = 20.1036
Hence the value of x is 20.1036 given that x³=8125.
Learn more about roots here:
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To solve the equation x^3 = 8125 for x, we need to take the cube root of both sides of the equation. The cube root of a number y, denoted as y^(1/3), is a number which, when raised to the power of 3, gives y.
So, let's calculate the cube root of 8125:
First, we can factor 8125 into its prime factors to help simplify the cube root:
8125 = 5 * 1625 = 5 * 5 * 325 = 5 * 5 * 5 * 65 = 5 * 5 * 5 * 5 * 13 (Since 65 = 5 * 13)
Now that we have the prime factorization of 8125, we can group the factors into triples:
8125 = (5 * 5 * 5) * 13
Notice that there are three 5s in one group and a leftover 13 that cannot form a triple.
The cube root of a product of factors is the product of the cube roots of those factors, so taking the cube root of both sides of the equation gives us:
x = (5 * 5 * 5)^(1/3) * 13^(1/3)
x = 5 * 13^(1/3)
Because 13 is a prime number and cannot be broken down any further, it does not have a rational cube root. As a result, 13^(1/3) is an irrational number and cannot be represented exactly as a fraction. However, we can write the exact answer as:
x = 5 * 13^(1/3)
To find an approximate rational representation, we can look for a fraction that is close to the cube root of 13. Since we cannot do this exactly without a calculator, we can only leave our final answer like this, with the understanding that the cube root of 13 remains irrational.
In conclusion, the exact result, in simplest form without using an irrational-to-rational conversion (which would not be exact), is:
x = 5 * 13^(1/3)
