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Exponents and Powers - Expressing numbers in Standard Form (Scientific Notation)

Grade 8ICSE

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

🔑Concepts

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Standard Form (Scientific Notation) is a method used to express very large or very small numbers in a concise way. It is written as the product of a decimal number between 1 and 10 and a power of 10. Visually, this replaces a long string of zeros with a single power of 10, making it easier to read and compare.

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The General Format: A number in standard form is written as a×10na \times 10^n, where 1≤a<101 \le a < 10. This means the coefficient 'aa' must have exactly one non-zero digit to the left of the decimal point. For example, 5.4×1035.4 \times 10^3 is standard, but 54×10254 \times 10^2 is not.

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Expressing Large Numbers: When converting a number larger than 10 to standard form, the decimal point moves to the left. The positive exponent nn represents the number of places the decimal point has shifted. Visually, imagine the decimal point jumping over digits from right to left until only one digit remains on its left side.

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Expressing Small Numbers: For numbers between 0 and 1, the decimal point moves to the right. The exponent nn becomes a negative integer, representing how many places the decimal shifted. Visually, you can see the decimal point sliding past leading zeros to sit immediately after the first non-zero digit.

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Comparison of Numbers: Standard form makes it easy to compare the magnitude of numbers. If two numbers are in standard form, the one with the higher power of 10 is larger. If the powers are the same, compare the coefficients 'aa'. For example, 2×1052 \times 10^5 is larger than 9×1049 \times 10^4 because the exponent 5 is greater than 4.

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Converting to Usual Form: To revert from standard form to usual form, look at the exponent nn. If nn is positive, move the decimal to the right by nn places. If nn is negative, move the decimal to the left by ∣n∣|n| places, adding placeholder zeros where necessary. This restores the number to its full expanded visual state.

📐Formulae

Standard Form: a×10na \times 10^{n}

Condition for Coefficient: 1≤a<101 \le a < 10

Condition for Exponent: n∈Zn \in \mathbb{Z} (where nn is an integer)

Negative Exponent Rule: 10−n=110n10^{-n} = \frac{1}{10^n}

Product Law: (a×10m)×(b×10n)=(a×b)×10m+n(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}

💡Examples

Problem 1:

Express the distance between the Earth and the Sun, approximately 149,600,000,000149,600,000,000 meters, in standard form.

Solution:

  1. Identify the current decimal point, which is at the end of the number: 149,600,000,000.0149,600,000,000.0
  2. Move the decimal point to the left until there is only one non-zero digit to its left.
  3. The decimal moves 11 places to the left to become 1.4961.496.
  4. Since we moved 11 places to the left, the exponent is +11+11.
  5. The standard form is 1.496×10111.496 \times 10^{11} meters.

Explanation:

To convert a large number, we count the number of shifts to the left to determine the positive power of 10.

Problem 2:

Express the size of a plant cell, which is 0.000012750.00001275 meters, in standard form.

Solution:

  1. Identify the first non-zero digit, which is 11.
  2. Move the decimal point to the right until it is placed after the 11: 1.2751.275
  3. Count the number of places the decimal moved: 1,2,3,4,51, 2, 3, 4, 5 places.
  4. Since the decimal moved to the right (making a small number larger), the exponent is negative: −5-5.
  5. The standard form is 1.275×10−51.275 \times 10^{-5} meters.

Explanation:

For decimals smaller than 1, we move the decimal point to the right and use a negative exponent to show how many places it was shifted.