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Scientific Notation And Powers Of 10

Represent very large or small numbers using expanded form with powers of 10, and standard Scientific Notation.

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Powers of 10 in Expanded Form

Writing numbers using powers of 10.

We often use numbers like 10, 100, and 1000 when writing Indian numerals in expanded form. For large numbers, we can simplify this by using powers of 10.

Take the number 47,561:
$47561 = (4 \times 10000) + (7 \times 1000) + (5 \times 100) + (6 \times 10) + 1$

Using powers of 10, this becomes:
$47561 = (4 \times 10^4) + (7 \times 10^3) + (5 \times 10^2) + (6 \times 10^1) + (1 \times 10^0)$

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Standard Form (Scientific Notation)

Defining the scientific notation format.

x \times 1 0^{y}

A method to concisely write very large or very small numbers.

Conditions

Coefficient ( $x$ ): Must be $1 \leq x < 10$ .
Exponent ( $y$ ): Must be any integer. It reflects the magnitude or number of digits.

Real-World Example

Distance from the Sun to Saturn:

$14, 33, 50, 00, 00, 000 m$
= $1.4335 \times 1 0^{12} m$

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Converting to Standard Form

Example of converting a large number.

Problem.
The distance between the Sun and Saturn is $14,33,50,00,00,000$ m. Express this enormous distance in standard scientific notation.

Step 1. Find the coefficient.
Place the decimal point immediately after the first non-zero digit so that the number is between $1$ and $10$ .
$x = 1.4335$

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Try It: Express in Standard Form

Faded practice for scientific notation.

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Let us practice expressing the large number 70,04,00,00,000 in standard form. Step 1 requires us to find the coefficient (x), such that (1 \le x < 10). By placing the decimal point right after the first digit (7), we get the coefficient . Step 2 involves counting the number of places the decimal point moved from the far right. Counting the digits we bypassed, we find that the decimal moved exactly 10 places. Therefore, the exponent (y) for our base 10 will be 10. Combining these pieces into the format (x \times 10^y), the final answer is (\times 10^{}).

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Real World Calculation

Applying scientific notation to a word problem.

The human body has about 38 trillion bacterial cells. Find the estimated bacterial population residing in all humans in the world if the global human population is 8.2 billion.

Use the step-by-step fields below to solve.

Recall: $1 \text{ trillion} = 10^{12}$ $1 \text{ billion} = 10^9$

Given values*

List the key values provided in the problem with their units.

Convert both to Scientific Notation*

Rewrite both quantities as a product of a number and a power of 10.

Multiply the quantities*

Set up the expression to find the total bacterial population.

Multiply coefficients and add exponents*

Show the calculation for multiplying the coefficients and combining the powers.

Final Standard Form*

Adjust the coefficient to be between 1 and 10 and correct the exponent.

Sanity Check (Optional)

Does the magnitude of your answer make sense for this context?

← previousDivision, Zero, And Negative Exponents next →Getting A Sense For Large Numbers