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For example, 3 5 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power. The word "raised" is usually omitted, and sometimes "power" as well, so 3 5 can be simply read "3 to the 5th", or "3 to the 5".
In arithmetic and algebra, the fifth power or sursolid [1] of a number n is the result of multiplying five instances of n together: n 5 = n × n × n × n × n. Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its cube. The sequence of fifth powers of integers is:
(Haynsworth inertia additivity formula) If A is invertible, then the inertia of the block matrix M is equal to the inertia of A plus the inertia of M/A. (Quotient identity) / = ((/) / (/)). [5] The Schur complement of a Laplacian matrix is also a Laplacian matrix. [6]
An interpolant () corresponds to a solution = (, …,) of the above matrix equation =. The matrix X on the left is a Vandermonde matrix , whose determinant is known to be det ( X ) = ∏ 1 ≤ i < j ≤ n ( x j − x i ) , {\displaystyle \textstyle \det(X)=\prod _{1\leq i<j\leq n}(x_{j}-x_{i}),} which is non-zero since the nodes x j ...
Zero to the power of zero, denoted as 0 0, is a mathematical expression that can take different values depending on the context. In certain areas of mathematics, such as combinatorics and algebra , 0 0 is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents .
To calculate a Pythagorean triple, take any term of this sequence and convert it to an improper fraction (for mixed number , the corresponding improper fraction is ). Then its numerator and denominator are the sides, b and a , of a right triangle, and the hypotenuse is b + 1 .
For the cases where has full row or column rank, and the inverse of the correlation matrix ( for with full row rank or for full column rank) is already known, the pseudoinverse for matrices related to can be computed by applying the Sherman–Morrison–Woodbury formula to update the inverse of the ...
The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product: = ⏟ = (( + )) = ( + ). For example, the first few powers of the imaginary unit i are i , i 2 = − 1 , i 3 = − i , i 4 = 1 , i 5 = i , … {\displaystyle i,i^{2}=-1,i^{3}=-i,i ...