### Exponents & More

Topics within Exponents & More

## Practice

In the previous section, we learned that powers are a shorthand way of writing repeated multiplication. Now, we can put our knowledge to the test by trying out some practice questions.

#### Part 1: Basic Properties

In this part, we will be exploring some interesting properties of exponents. Type answers in exponenetial form.

##### Example 1.1:

$\displaystyle { 2^3\cdot2^4=?}$ $$2^3\cdot2^4 \\~\\ =(2\cdot2\cdot2)\cdot(2\cdot2\cdot2\cdot2) \\~\\ =2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2 \\~\\ =2^7$$

##### Example 1.2:

$\displaystyle { 2^4\cdot3^4=?}$ $$2^4\cdot3^4~\\~\\ =\left(2\cdot2\cdot2\cdot2\right)\cdot\left(3\cdot3\cdot3\cdot3\right)~\\~\\ =\left(2\cdot3\right)\cdot\left(2\cdot3\right)\cdot\left(2\cdot3\right)\cdot\left(2\cdot3\right)~\\~\\ =\left(2\cdot3\right)^4~\\~\\ =6^4$$

• $\displaystyle{ \frac{2^7}{2^3}= }$

Write out all the 2's being multiplied.
How can we simplify the fraction?
$$\frac{2^7}{2^3}~\\~\\ =\frac{2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2}{2\cdot2\cdot2}~\\~\\ =2\cdot2\cdot2\cdot2~\\~\\ =2^4$$
• $\displaystyle{ \frac{6^4}{3^4}= }$

Write out $6^4$ and $3^4$.
How can we regroup the factors?
$$\frac{6^4}{3^4}~\\~\\ =\frac{6\cdot6\cdot6\cdot6}{3\cdot3\cdot3\cdot3}~\\~\\ =\left(\frac{6}{3}\right)\cdot\left(\frac{6}{3}\right)\cdot\left(\frac{6}{3}\right)\cdot\left(\frac{6}{3}\right)~\\~\\ =\left(\frac{6}{3}\right)^4~\\~\\ =2^4$$
• $\displaystyle{ \left(5^2\right)^4= }$

Write out $5^2$.
How many 5's are being multiplied together?
$$\left(5^2\right)^4~\\~\\ =\left(5\cdot5\right)^4~\\~\\ =\left(5\cdot5\right)\cdot\left(5\cdot5\right)\cdot\left(5\cdot5\right)\cdot\left(5\cdot5\right)~\\~\\ =5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5~\\~\\ =5^8$$

• #### Part 2: Powers of Ten

Here, we'll find patterns by exploring powers of ten. Type answers in simplest form.

##### Example 2.1:

$\displaystyle{ 10^2=? }$ $$10^2~\\~\\ =10\cdot10~\\~\\ =100$$

• $\displaystyle{ 10^3= }$

How many 10's are being multiplied together?
$$10^3~\\~\\ =10\cdot10\cdot10~\\~\\ =1000$$
• $\displaystyle{ 10^4= }$

How many 10's are being multiplied together?
$$10^4~\\~\\ =10\cdot10\cdot10\cdot10~\\~\\ =10000$$

• #### Part 3: Interesting Exponents

In this part, we'll look at what happens when the exponent is 1 or 0. Type answers in simplest form.

• $\displaystyle{ 10^1= }$

How many 10's are being multiplied together?
How many zeroes does $10^2$ have when written as an integer? What about $10^3$ and $10^4$? How many zeroes do you think $10^1$ has?

1. Recall that when we write out exponents using multiplication, we can start by multiplying by 1. In $10^1$, there is one 10 being multiplied. $$1\cdot 10=10$$ Therefore, the answer is 10.
2. Notice that $10^x$ ends with $x$ zeroes. $$10^4=10000~\\~\\ 10^3=1000~\\~\\ 10^2=100$$ So, $$10^1=10$$
3. We know that $$\frac{x^a}{x^b}=x^{a-b}$$ (See #1 in Part 1). Therefore, $$\frac{10^3}{10^2}=10^{3-2}=10^1$$ We also know that $$\frac{10^3}{10^2}=\frac{1000}{100}=10$$ So, $$10^1=10$$
• $\displaystyle{ 10^0= }$

How many 10's are being multiplied together?
How many zeroes do $10^1$, $10^2$, $10^3$, and $10^4$ have? How many zeroes do you think $10^0$ has?

2. When $x$ is positive, $10^x$ is a 1 followed by $x$ zeroes when written as an integer. Continuing the pattern, $10^0$ is a 1 followed by 0 zeroes, which is just 1.
3. We know that $$\frac{x^a}{x^b}=x^{a-b}$$ (See #1 in Part 1). Therefore, $$\frac{10^1}{10^1}=10^{1-1}=10^0$$ We also know that $$\frac{10^1}{10^1}=10/10=1$$ So, $$10^0=1$$