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