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:

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

  • $\displaystyle{ \frac{2^7}{2^3}= }$

    Hint Write out all the 2's being multiplied.
    Hint How can we simplify the fraction?
    Show/Hide Solution $$\frac{2^7}{2^3}~\\~\\ =\frac{2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2}{2\cdot2\cdot2}~\\~\\ =2\cdot2\cdot2\cdot2~\\~\\ =2^4$$

  • $\displaystyle{ \frac{6^4}{3^4}= }$

    Hint Write out $6^4$ and $3^4$.
    Hint How can we regroup the factors?
    Show/Hide Solution $$\frac{6^4}{3^4}~\\~\\ =\frac{6\cdot6\cdot6\cdot6}{3\cdot3\cdot3\cdot3}~\\~\\ =\left(\frac{6}{3}\right)\cdot\left(\frac{6}{3}\right)\cdot\left(\frac{6}{3}\right)\cdot\left(\frac{6}{3}\right)~\\~\\ =\left(\frac{6}{3}\right)^4~\\~\\ =2^4$$

  • $\displaystyle{ \left(5^2\right)^4= }$

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

    • Part 2: Powers of Ten

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

    Example 2.1:

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

  • $\displaystyle{ 10^3= }$

    Hint How many 10's are being multiplied together?
    Show/Hide Solution $$10^3~\\~\\ =10\cdot10\cdot10~\\~\\ =1000$$

  • $\displaystyle{ 10^4= }$

    Hint How many 10's are being multiplied together?
    Show/Hide Solution $$10^4~\\~\\ =10\cdot10\cdot10\cdot10~\\~\\ =10000$$

    • 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.

  • $\displaystyle{ 10^1= }$

    Hint How many 10's are being multiplied together?
    Hint 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?
    Show/Hide Solution

    Here are several ways to think about this:

    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, $$10^1=10$$

  • $\displaystyle{ 10^0= }$

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

    Here are several ways to think about this:

    1. Recall that when we write out exponents using multiplication, we can start by multiplying by 1. So if there is nothing being multiplied, the only factor is 1, so the answer is 1.
    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, $$10^0=1$$