# Everything I know about Exponents

1) Represent repeated multiplication with exponents

3x3x3x3 = 81         $3^4$ = 81

2) Describe how powers represent repeated multiplication

The power above is $3^4$. Powers have a base and an exponent. The base (3) would represent the number multiplied by the exponent (4) as shown in number one. The exponent is how many times 3 would be multiplied. This power would be the equivalent to 3x3x3x3.

3) Demonstrate the difference between the exponent and the base by building models of a given power, such as $2^3$ and $3^2$

4) Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated mulitplication, such as $2^3$ and $3^2$

$2^3$ = 2x2x2 = 8

$3^2$ = 3×3 = 9

5) Evaluate powers with integral bases (excluding base 0) and whole number exponents.

$({-2})^4$ = (-2)(-2)(-2)(-2) = 16

$5^3$ = 5x5x5 = 125

$6^5$ = 6x6x6x6x6 = 7,776

$({-4})^5$ = (-4)(-4)(-4)(-4)(-4) = -1024

6) Expain the role of parentheses in powers by evaluating a given set of powers such as $({-2})^4$ , $({-2}^4)$$-2^4$

With $({-2})^4$ , the repeated multiplication would be (-2)(-2)(-2)(-2) which would equal 16.

With $({-2}^4)$ , You are applying the exponent of 4 to the base of 2, then you will apply the negative symbol in the form of -1.

-1x2x2x2x2 = -16

With ${-2}^4$ , You are essentially doing the same thing as $({-2}^4)$.

-1x2x2x2x2 = -16

7) Explain the exponent laws for multiplying and dividing powers with the same base.

(Product Law)

When multiplying powers with the same base,  keep the base, then add the exponents. For example, $3^4$ x $3^5$ = $3^9$ = 19,683

(Quotient Law)

When diving powers with the same base, you keep the base, then subtract the exponents. For example,

$3^8$ ÷ $3^3$ = $3^5$ = 243

8) Explain the exponent laws for raising a product and quotient to an exponent.

(Power Law)

When raising a power to an exponent you keep the base, multiply the exponent(s), and if there is a coefficient, apply the exponent to the coefficient.

For example,

2 * $(5^3)^2$ = $2^2$ * $5^6$ = 4*15,625 = 62,500

9) Explain the law for powers with an exponent of zero.

Any base (except 0) with an exponent of zero will equal 1.

$4^0$ = 1

$0^0$ = 0

10) Use patterns to show that a power with an exponent of zero is equal to one.

$4^5$ = 1024

$4^4$ = 256

$4^3$ = 64

$4^2$ = 16

$4^1$ = 4

$4^0$ = 1

The base (4) with exponent (0) will essentially have 4 divide into itself, so 4  ÷  4 = 1

11) Explain the law for powers with negative exponents.

When dealing with negative exponents, any base (except zero) raised to a negative exponent will equal the reciprocal of the base raised to a positive exponent, for example:

$4^{-3}$ = $\frac {4^{-3}}{1}$ = $\frac {1}{4^3}$

12) Use patterns to explain the negative exponent law.

$3^5$ = 243

$3^4$ = 81

$3^3$ = 27

$3^2$ = 9

$3^1$ = 3

$3^0$ = 1

$3^{-1}$ = $\frac{1}{3^1}$

13) I can apply the exponent laws to powers with both integral and variable bases.

Yes, I can.

$x^5$ * $x^4$ = $x^9$

$5^8$ ÷$5^5$ = $5^3$ = 125

$(2^2)^4$  = $2^6$ = 64

$x^0$ = 1

14) I can identify the error in a simplification of an expression involving powers.

Yes,

$(2a^3b^2)^3$ = $6a^9b^6$

$(2a^3b^2)^3$ = $8a^9b^6$

When simplifying a power using the power rule, you have to remember to apply the coefficient to the exponent not mulitply them together. With the first example above, the coefficient was multiplied by 3 rather than applied the exponent of 3. This made it equal 6 rather than 8, which is incorrect.

15) Use the order of operations on expressions with powers.

$(5x^3)^2$ x $(2x^5)^4$ = $5^2x^6$ x $2^4x^5$ = 25*16$x^{11}$ = 400$x^{11}$

I first did the power law which states that the exponents are applied to the coefficients, and the powers are raised by the exponent. Then, I did the product law which multiples the coefficients and adds the exponents of the powers of the same base.

16) Determine the sum and difference of two powers.

$3^4$ + $6^3$ = 81 + 216 = 297

$3^8$$4^4$ = 6,561 – 256 = 6,305

17) Identify the error in applying the order of operations in an incorrect solution

In equations like this, you would do the power law first. However, the negative exponent law was done first, here is the result.

$(x^{-3}y^4z^3)^2$ x $(x^8y^3z^6)$ = $\frac{y^4z^3}{x^3}$

This is going to take much longer to find the answer. However, you can do the power law first.  So,

$(x^{-6}y^8z^6)$ x $(x^8y^3z^6)$ = $(x^2y^11z^12)$

18) Use powers to solve problems (measurement problems)

The side length of a cube is 5 cm, what is the volume of the cube?

Well, in order to find the volume of the cube, you have to find the length, the width, and the height, since this is a cube, they are all going to be equal so you can find the volume of the cube by doing this:

$5^3$ = 125 $cm^3$

19) Use powers to solve problems (growth problems)

A bacterium quadruples itself in 1 hour, How many will there be in 5 hours?

The base represents how many bacteria are produced, and the exponent represents time (5 hours).

$4^5$

20) Applying the order of operations on expressions with powers involving negative exponents and variable bases.

1. $(3x^{-2}y^7) ^2$ ÷ $(3x^{-4}y^8)^4$ = $(3^2x^{-4}y^14)$  ÷ $3^4x^{16}y^{32}$ = $(3^{-2}x^{-20}y^{-18})$ = $\frac{1}{3^2x^{20}y{18}}$

# Principles of Flight

Tumble Gliders

Bernoulli’s principle states that: Low Velocity = High Pressure and High Veolocity = Low pressure. If you imagine an airplane wing, It is designed for the air on the top of the wing to be faster than that of the bottom of the wing. Since the air is slower on the bottom it creates a high pressure zone which will create lift.

Making Rockets

1. Draw a path of trajectory of your rocket.

2. Which force is acting on the rocket at the moment of launch? (use arrows to indicate direction)

Thrust

3. As the rocket was half-way up, which force(s) is/are acting on the rocket? (use arrows)

Drag, Weight.

4. As the rocket begins to veer into another direction, which force is acting on the rocket? Explain why this is happening.

Lift, Bernoulli’s principle states that: Low Velocity = High Pressure and High Velocity = Low Pressure. The air going over the rocket is travelling at a faster rate than the air under the rocket. The high pressure on the bottom will create lift and keep the rocket up.

5. Did some rockets work better than others? How does the shape of the nose and fin effect the trajectory of the rocket? Explain in terms of the four forces that act on a rocket ship.

Yes, some did. I found that if the nose better fits the fuselage, the farther it went. Without proper fins, the rocket just sat in the air and spun on the way down. The main goal of making your rocket go far is that you decrease drag while increasing lift. If the nose is flat on top of your rocket it will increase the amount of drag on your rocket making it not go so high. If it is shaped in a way where it cuts through the air, the rocket will be more aerodynamic and fly much farther. Same for the fins, the fins are also supposed to make your rocket more aerodyamic and decrease drag, If they are positioned in a way that makes it cut through the air, the fins will also contribute to less drag.

Water Rocket

Vocabulary:

Acceleration: The rate that an object increases in speed.

Center of Drag: The center of which drag will act upon an object the most.

Center of Mass: The center of which mass of an object will balance.

Drag: An accumulative amount of a resistant force on an object.

Inertia: An object will be in rest unless acted upon by an outside force.

Mass: A unit of measurement relative to an objects property.

Momentum: The amount of movement an object has.

Pressure: A force that exerts force upon another object. (Gas,Liquid, Solid).

Velocity: A measurment of how fast something will move in  a certain direction.

1. How did the height you estimated your rocket would reach compare with the actual estimated height?

To be honest, I thought it would’ve gone a little bit higher than it did.

2. What do you think might have caused any differences in the height you achieved?

Shape of the cone, shape and positions of the fins.

3. Did your rocket launch straight up? If not, why do you think it veered off course?

No, it initally veered off course. I think it was because of the top of my rocket. The top of the rocket was slightly crooked and did not point up straight. The cone was also more slightly loose, but it helped with the deployment of both of my parachutes.

4. Do you think that this activity was more rewarding to do as a team, or would you have
preferred to work alone on it? Why?

I would preferred everyone did their own rocket. I think it is a more satisfying experience if the rocket was sucessful because it was your work and not between two people. Work would be split between two people which can make it harder to agree on a design and bring two seperate aspects of the rocket together.

5. Did you adjust your model rocket at all? How? Do you think this helped or hindered

At first, the size of the fins on my rocket were going to be much bigger than they were. But, if they were the size I intended them to be, they wouldn’t fit very well on the rocket and be much more blocky. I think making the fins smaller and more sleek along the rocket actually helped my rocket.

6. How do you think the rocket would have behaved differently if it were launched in a weightless atmosphere?

If it was in a weightless atmosphere, I think the rocket would go farther, but the speed it would go be much more slowed down.

7. What safety measures do you think engineers consider when launching a real rocket?
Consider the location of most launch sites as part of your answer.

Making sure the area around the lauch pad is clear of debris. Building the launch pad on relatively flat terrain which is clear of any trees and such. Making sure that the rocket is clear for takeoff if  there was any air traffic up in the sky.

8. When engineers are designing a rocket which will carry people in addition to cargo, how do you think the rocket will change in terms of structural design, functionality, and features?

Rocket design would accommodate living quarters for those on the rocket. To increase ability to communicate with those who are on land. Depending on the length of travel or if they are orbiting around Earth, the rocket would not only need space for food, water, and toiletries, but something would have to send it to those in the rocket.

9. Do you think rocket designs will change a great deal over the next ten years? How? I think designs for rockets will have little change over the next ten years. Maybe a few moderate changes. The design of a rocket could be made to accommodate more people, be slightly more aerodynamic, or become more fuel efficient.

10. What tradeoffs do engineers have to make when considering the space/weight of fuel vs. the weight of cargo?

The more fuel there is, the more the rocket will weigh and there would be less space for the cargo. If there is a lot of cargo, the rocket will weigh more and there wouldn’t be sufficient space for fuel in order to get to a set  destination especially if it is a good distance away from Earth.