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Exponents and Powers - Use of Exponents to Express Small Numbers in Standard Form

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Standard Form Definition: A small number is expressed in standard form as k×10nk \times 10^{n}, where 1≤k<101 \le k < 10 and nn is a negative integer. Visualize this as converting a long decimal with many leading zeros into a more compact multiplication expression.

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Decimal Point Movement: To convert a decimal less than 11 to standard form, the decimal point must move to the right until it is placed immediately after the first non-zero digit. Imagine the decimal point 'jumping' over each zero, where each jump represents an increase in the negative power of 1010.

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Negative Exponents: Small numbers use negative exponents because they represent division by powers of 1010. For example, 10−310^{-3} is the same as 1103\frac{1}{10^{3}} or 11000\frac{1}{1000}, which equals 0.0010.001.

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Counting the Shift: The value of the negative exponent nn is determined by the number of places the decimal point shifted to the right. If the decimal moves 55 places to the right to create a number between 11 and 1010, the power is 10−510^{-5}.

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Comparing Small Numbers: To compare two small numbers in standard form, first check the exponents. The number with the greater exponent (the one closer to zero) is the larger number. For instance, 10−410^{-4} is larger than 10−710^{-7}. Visualize this on a number line where 10−410^{-4} is further to the right than 10−710^{-7}.

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Conversion back to Usual Form: To convert k×10−nk \times 10^{-n} back to usual decimal form, move the decimal point nn places to the left, adding zeros as placeholders where necessary. This restores the number to its original 'small' decimal appearance.

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Scientific Scale: Standard form is used to represent physical constants and microscopic measurements, such as the size of a plant cell (roughly 0.000012750.00001275 m), making it easier to read and perform calculations without losing track of zeros.

📐Formulae

k×10n, where 1≤k<10 and n∈Zk \times 10^{n}, \text{ where } 1 \le k < 10 \text{ and } n \in \mathbb{Z}

a−n=1ana^{-n} = \frac{1}{a^{n}}

10−n=110n=0.00…0⏟n−1110^{-n} = \frac{1}{10^{n}} = 0.\underbrace{00\dots0}_{n-1}1

Decimal 0.00…0⏟mx=x×10−(m+1)\text{Decimal } 0.\underbrace{00\dots0}_{m}x = x \times 10^{-(m+1)}

💡Examples

Problem 1:

Express the number 0.00000000000850.0000000000085 in standard form.

Solution:

Step 1: Identify the first non-zero digit, which is 88.\nStep 2: Move the decimal point to the right until it is after 88. Count the number of places it moves.\nStep 3: The decimal moves 1212 places to the right to become 8.58.5.\nStep 4: Since the decimal moved to the right by 1212 places, the exponent of 1010 will be −12-12.\nStep 5: Write the result as 8.5×10−128.5 \times 10^{-12}.

Explanation:

To convert a small decimal, we move the decimal point until we have a coefficient between 11 and 1010. The count of moves becomes the negative exponent.

Problem 2:

Express 0.00003240.0000324 in standard form.

Solution:

Step 1: Move the decimal point to the right to get a number between 11 and 1010. Moving it 55 places gives 3.243.24.\nStep 2: Because we moved the decimal point 55 places to the right, the power of 1010 is −5-5.\nStep 3: Combine the coefficient and the power: 3.24×10−53.24 \times 10^{-5}.

Explanation:

This conversion follows the rule k×10nk \times 10^{n} where k=3.24k = 3.24 and n=−5n = -5 because the decimal shifted 55 positions to the right.