To simplify the power of a product of two exponential expressions, we can use the
power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider
$\text{\hspace{0.17em}}{\left(pq\right)}^{3}.\text{\hspace{0.17em}}$ We begin by using the associative and commutative properties of multiplication to regroup the factors.
In other words,
$\text{\hspace{0.17em}}{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}.$
The power of a product rule of exponents
For any real numbers
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and any integer
$\text{\hspace{0.17em}}n,$ the power of a product rule of exponents states that
${\left(ab\right)}^{n}={a}^{n}{b}^{n}$
Using the power of a product rule
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
${\left(a{b}^{2}\right)}^{3}$
${\left(2t\right)}^{15}$
${\left(\mathrm{-2}{w}^{3}\right)}^{3}$
$\frac{1}{{\left(\mathrm{-7}z\right)}^{4}}$
${\left({e}^{\mathrm{-2}}{f}^{2}\right)}^{7}$
Use the product and quotient rules and the new definitions to simplify each expression.
To simplify the power of a quotient of two expressions, we can use the
power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
For any real numbers
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and any integer
$\text{\hspace{0.17em}}n,$ the power of a quotient rule of exponents states that
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.