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
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0
then
4x = 2-3
4x = -1
x = -(1÷4) is the answer.
Jacob
4x-2+3
4x=-3+2
4×=-1
4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3
4x=-3+2
4x=-1
4x÷4=-1÷4
x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?