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We often find ourselves in situations where we need to perform repeated multiplication of the same number. Such mathematical expressions can be simplified using the concept of powers and exponents. Since we are dealing with repeating multiplication, we can figure out a pattern among the products. We will look at the pattern among the powers of 10. ...Read MoreRead Less
A power expression is an expression that represents repeated multiplication of the same factor and can be represented as \(x^{n}\) . Simply put, a power is the result of repetitive multiplication of the same number. The repeated number is called a factor or is the base.
The number of times the base is used as a factor is given by the exponent of a power. In a place value chart, every time you multiply a number by 10, it moves one position to the left. The number of zeroes in the product equals the exponent.
When two powers have the same base, they can be multiplied. We add the exponents of two powers when multiplying them.
When you look at an exponential function, you’ll notice the larger number on the bottom and a small number next to it in the upper right-hand corner. The “base” is the large number at the bottom, and the “exponent” is the small number in the corner, and this is written as a superscript.
For example, 10 x 10 x 10 x 10 = \(10^{4}\)
Here, the number 10 is used as a factor 4 times. So, the exponent is 4.
And finally, the power is \(10^{4}\).
When you find powers of a number in order, the list of products that results, forms a pattern. We can predict the next product on the list by looking at this pattern. The standard form refers to any number that can be written as a decimal number between 1 and 10, multiplied by a power of 10. A number is written in standard form to make it easier to read. It’s frequently used for extremely large or extremely small numbers.
With a single step of multiplication, we can find the standard form of a number raised to the nth power and the standard form of that number raised to the n+1 power. When you find powers of 10 in a row, the resulting list of products has a pattern of zeros.
There are several patterns to be found in the table below:
Exponential Form (Power) | Standard Form | Description |
---|---|---|
\(10^{0}\) | 1 | The power 10 has exponent 0; the number 1 has no zeros. |
\(10^{1}\) | 10 | The power 10 has exponent 1; the number 10 has 1 zero. |
\(10^{2}\) | 100 | The power 10 has exponent 2; the number 100 has 2 zeros. |
\(10^{3}\) | 1,000 | The power 10 has exponent 3; the number 1,000 has 3 zeros. |
\(10^{4}\) | 10,000 | The power 10 has exponent 4; the number 10,000 has 4 zeros. |
\(10^{5}\) | 1,00,000 | The power 10 has exponent 5; the number 1,00,000 has 5 zeros. |
\(10^{6}\) | 1,000,000 | The power 10 has exponent 6; the number 1,000,000 has 6 zeros. |
\(10^{7}\) | 10,000,000 | The power 10 has exponent 7; the number 10,000,000 has 7 zeros. |
\(10^{8}\) | 100,000,000 | The power 10 has exponent 8; the number 100,000,000 has 8 zeros. |
\(10^{9}\) | 1000,000,000 | The power 10 has exponent 9; the number 1,000,000,000 has 9 zeros. |
For example, To find the value of 3 x \(10^{4}\).
We have to multiply 3 by the powers of 10. Look at the pattern below: Notice the pattern that in each product the number of zeros after 3 is equal to the exponent.
Exponential Form (Power) | Standard Form | Description |
---|---|---|
3 x \(10^{1}\) | 3 x 10 = 30 | The power 10 has exponent 1; the number 10 has one zero. |
3 x \(10^{2}\) | 3 x 100 = 300 | The power 10 has exponent 2; the number 100 has 2 zeros. |
3 x \(10^{3}\) | 3 x 1000 = 3000 | The power 10 has exponent 3; the number 1,000 has 3 zeros. |
3 x \(10^{4}\) | 3 x 10,000 = 30,000 | The power 10 has exponent 4; the number 10,000 has 4 zeros. |
In math, powers are useful because they allow us to shorten a number that would otherwise be extremely tedious to write. For example, if we want to express the product of “A” multiplied by itself 6 times, without powers we’d only be able to write this as, A x A x A x A x A x A, which is A multiplied by itself 6 times in a row. So we need a different way to write this expression, and that’s what powers are used for. Instead of writing out A multiplied by itself 6 times, we can just write \(A^{6}\) .
1) Write the product of 10 x 10 x 10 x 10 x 10 x 10 as a power.
Solution: 10 x 10 x 10 x 10 x 10 x 10 =\(10^{6}\)
Here, the number 10 is used as a factor for 6 times. So, the exponent is 6.
And finally, 10 x 10 x 10 x 10 x 10 x 10 can be written as \(10^{6}\)
2) Find the value of 5 X \(10^{4}\).
Solution: We have to multiply 5 by the powers of 10. Look at the pattern below and notice the pattern in each product, along with the number of zeros after 5, which is equal to the exponent.
Exponential Form (Power) | Standard Form | Description |
---|---|---|
5 x \(10^{1}\) | 5 x 10 = 50 | The power 10 has exponent 1; the number 10 has one zero. |
5 x \(10^{2}\) | 5 x 100 = 500 | The power 10 has exponent 2; the number 100 has 2 zeros. |
5 x \(10^{3}\) | 5 x 1000 = 5000 | The power 10 has exponent 3; the number 1,000 has 3 zeros. |
5 x \(10^{4}\) | 5 x 10,000 = 50,000 | The power 10 has exponent 4; the number 10,000 has 4 zeros. |
3) Find the value of 6 x \(10^{3}\).
Solution: We have to multiply 6 by the powers of 10. Look at the pattern below and check how in the pattern, and in each product the number of zeros after 6 is equal to the exponent.
Exponential Form (Power) | Standard Form | Description |
---|---|---|
6 x \(10^{1}\) | 6 x 10 = 60 | The power 10 has exponent 1; the number 10 has one zero. |
6 x \(10^{2}\) | 6 x 100 = 600 | The power 10 has exponent 2; the number 100 has 2 zeros. |
6 x \(10^{3}\) | 6 x 1000 = 6000 | The power 10 has exponent 3; the number 1,000 has 3 zeros. |
4) Write the number in expanded form using exponents?
a) 63124
b) 862
Solution: 63124 = (6 x \(10^{4}\)) + (3 x \(10^{3}\)) + (1 x \(10^{2}\)) + 2 x \(10^{1}\) + 4
962 = (9 x \(10^{2}\)) + (6 x \(10^{1}\)) + 2
5) John and Joseph are contending for the position of mayor. What was the voter participation in the election? Calculate the total number of votes cast for each candidate.
John – \(10^{6}\) = ?
Joseph – 7 x \(10^{4}\) = ?
Also add the votes for John and Joseph.
Solution: John – \(10^{6}\) = 1,000,000
Exponential Form (Power) | Standard Form | Description |
---|---|---|
\(10^{6}\) | 1,00,000 | The power 10 has exponent 6; the number 1,00,000 has 6 zero. |
Joseph – 7 x \(10^{4}\) = 70,000
Exponential Form (Power) | Standard Form | Description |
---|---|---|
7 x \(10^{4}\) | 7 x 10,000 = 70,000 | The power 10 has exponent 4; the number 10,000 has 4 zeros. |
Adding the votes of John and Joseph = 1,000,000 + 70,000 = 1,070,000 people voted in the election.
6) For the past two years, a grocery store has been open. What are the total sales for both the first and second years?
Year | Sales |
---|---|
1 | \(10^{5}\) $ |
2 | 5 x \(10^{5}\) $ |
Solution: Year 1 sales = \(10^{5}\) $ = 1,00,000 $
Exponential Form (Power) | Standard Form | Description |
---|---|---|
\(10^{5}\) | 1,00,000 | The power 10 has exponent 5; the number 1,00,000 has 5 zeros. |
Year 2 sales = 5 \(10^{5}\) = 5,00,000 $
Exponential Form (Power) | Standard Form | Description |
---|---|---|
5 x \(10^{5}\) | 5 x 1,00,000 = 5,00,000 | The power 10 has exponent 5; the number 1,00,000 has 5 zeros. |
Total sales of year 1 and year 2 = 1,00,000 + 5,00,000
= 6,00,000 $
7) On which day did the event gather the most people? How many people are there?
Day | Event attendance |
---|---|
Friday | 10 x 10 |
Saturday | 5 x \(10^{2}\) |
Solution: Friday = 10 x 10 = 100
Saturday = 5 x \(10^{2}\) = 5 x 10 x 10 = 500
Total number of people = 100 + 500 = 600
In math, powers are useful because they allow us to shorten something that would otherwise be take time to write. For example, instead of writing out G multiplied by itself 12 times, we can write G¹².
Patterns are logically repeating elements, such as vertical stripes on a sweater. They can take the form of numbers, images, or shapes. Patterns assist children in making predictions by allowing them to anticipate what will happen next. They also assist children in learning how to reason and make logical connections. Patterns can be found in everyday life and should be pointed out to young children.