计值法

Expanded Notation Example Problems

We're going to dive into some types of problems that you can expect to see when working with expanded notation. First, if you haven't already, go check out this explanation on expanded notation.

Example 1: Putting a Number Into Expanded Notation

You'll see some problems that ask you to expand a given number.

We have a $3$ in the thousands place, $2$ in the hundreds place, $9$ in the tens place, and a $4$ in the ones place. This means we can add $3{,}000+200+90+4$ to get $3{,}294$.

Example 2: Condensing Expanded Notation Into a Number

Some of these problems will have you do the reverse where you have to condense an expanded number.

We can compress this by either adding the numbers together, or by taking the non-zero numbers from each term. $${\color{#29A603}{4}}{,}000+{\color{#29A603}{5}}00+{\color{#29A603}{1}}0+{\color{#29A603}{9}}=4{,}519$$

Example 3: Breaking Down Expanded Notation

These will make sure you understand the underlying theory of expanded notation.

This version of expanded notation shows each term as a digit times a power of $10$. For instance, we could write $300$ as $3 \times 100$. We are looking for $\bigcirc$, which corresponds to our hundreds place. This means that $\bigcirc$ could only be $9$ from our given options.

Example 4: Expanded Notation with Exponents

Similar to the last problem, these problems will have you break down expanded notation, but they'll also give you practice on exponents.

Let's brush up on our exponents of $10$.

• $10^0=1$
• $10^1=10$
• $10^2=100$
• $10^3=1{,}000$

We can multiply our digits by each of these exponents to find the expanded form. $9$ in the thousands place is $9\times 10^3$. $2$ in the hundreds place is $2\times10^2$. $4$ in the tens place is $4\times10^1$. $1$ in the ones place is $1\times10^0$. Combined, we get our expanded notation of $9 \times 10^3 + 2\times 10^2+ 4\times10^1+ 1\times10^0$.

Wrap-Up

Expanded notation can be attacked from many different angles, but they all come down to know what each part of the notation means.

For the expanded form of a whole number, simply write out each digit with the number of zeroes corresponding to is place value. This is in contrast to scientific notation, which would indicate the number of digits.

Example: Write $49{,}183$ in expanded notation.

As we see in the image, the answer is simply $40{,}000+9{,}000+100+80+3$. We can write these numbers in any order. That is because of the commutative property of addition.

Using exponents, this can also be rewritten as $$4\cdot 10^4+9\cdot 10^3+1\cdot 10^2+8\cdot 10^1+3\cdot 10^0.$$

(Video) Expanded Notation

by GreeneMath.com

Here's a video by GreeneMath.com that explains expanded notation.

$\large{\text{Summary}}$

Expanded notation is just writing a number as multiples of $1$'s, $10$'s, $100$'s etc. You take the digits and multiply them by their place value.

Let's look at an example. Write $675$ in expanded notation.

We take the first digit, $6$, and multiply it by its place value, hundreds. $$6\times100$$ Then we move onto the next digit and do the same, adding it to the previous. $$6\times100+7\times10+5\times1$$

Let's look at another example: $53{,}002$. We have $5$ in the ten thousands place, $3$ in the thousands place, and $2$ in the ones place. We can ignore the zeros since zero times any number is just zero. $$5\times10{,}000+3\times1{,}000+2\times1$$

Here's a quick way to write a number in expanded form. This is especially for you visual learners out there.

First, write down your number. In this example, I'll use $4{,}832$. $$4{,}832$$

Next, move each digital straight down to its own line. $$\begin{array}{cccc} 4& & & \\ &8& & \\ & &3& \\ & & &2 \end{array}$$

Fill in the upper corner with zeros. $$\begin{array}{cccc} 4&\color{#29A603}{0}&\color{#29A603}{0}&\color{#29A603}{0}\\ &8&\color{#29A603}{0}&\color{#29A603}{0}\\ & &3&\color{#29A603}{0}\\ & & &2 \end{array}$$

Finally, put +'s at the end of each line except the last. $$\begin{array}{ccccc} 4&0&0&0&+\\ &8&0&0&+\\ & &3&0&+\\ & & &2& \end{array}$$

Writing this on a single line is your expanded notation! $$4{,}000+800+30+2$$