Solve the problem 132 base (n)=72 base ten
To solve this problem, we need to find the value of n.
In base n, the number 132 is equal to 1*n^2 + 3*n^1 + 2*n^0.
In base ten, the number 72 is equal to 7*10^1 + 2*10^0.
Setting these two expressions equal, we get:
1*n^2 + 3*n^1 + 2*n^0 = 7*10^1 + 2*10^0
Re-arranging the equation, we have:
1*n^2 + 3*n - 68 = 0
To solve for n, we can use the quadratic formula:
n = (-3 ± √(3^2 - 4*1*(-68))) / (2*1)
n = (-3 ± √(9 + 272)) / 2
n = (-3 ± √281) / 2
Therefore, the value of n is approximately -9.36 or 6.36, since n cannot be negative in this case.
To solve the problem, we need to find the value of the base "n" in order to convert the number 132 from base "n" to base ten (decimal).
To convert a number from base "n" to base ten, we can use the expanded form method. For example, the number 132 in base 10 can be expressed as: (1 * n^2) + (3 * n^1) + (2 * n^0).
Given that 132 base (n) is equal to 72 in base ten, we can rewrite the equation as follows:
(1 * n^2) + (3 * n^1) + (2 * n^0) = 72
Simplifying the equation, we have:
n^2 + 3n + 2 = 72
Rearranging the equation, we get:
n^2 + 3n - 70 = 0
Now, we can solve this quadratic equation by factoring or by using the quadratic formula.
Factoring:
(n + 10)(n - 7) = 0
Setting each factor to zero and solving for "n", we obtain two possible values for the base:
n + 10 = 0 --> n = -10
n - 7 = 0 --> n = 7
However, we cannot have a negative base in this context, so the valid solution is:
n = 7
Hence, 132 base 7 is equal to 72 base ten.
To solve the problem of converting the number 132 in base (n) to base 10, we need to determine the value of the unknown base, n.
Let's break down 132 in base (n) into its individual digits. The rightmost digit is the one's place, the next digit to the left is the n's place, and so on.
Using the base (n) representation, we can write 132 as:
(1 * n^2) + (3 * n^1) + (2 * n^0)
Now, in base 10, 72 can be expressed as:
(7 * 10^1) + (2 * 10^0)
We can equate the two expressions and solve for n:
(1 * n^2) + (3 * n^1) + (2 * n^0) = (7 * 10^1) + (2 * 10^0)
Rearranging terms, we get:
n^2 + 3n + 2 = 70 + 2
Simplifying:
n^2 + 3n + 2 = 72
Rearranging and subtracting 72 from both sides:
n^2 + 3n - 70 = 0
We have now transformed the problem into a quadratic equation. To solve it, we can use factoring, completing the square, or using the quadratic formula.
Factoring the quadratic equation:
(n + 10)(n - 7) = 0
Setting each factor to zero and solving for n:
n + 10 = 0 --> n = -10
n - 7 = 0 --> n = 7
Since we are dealing with a base, n must be a positive integer. Therefore, n = 7 is the valid solution.
So, the base (n) is 7.