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Power Play - A Pinch of History

Grade 8CBSE

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

🔑Concepts

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Introduction to Exponents: When a number is multiplied by itself repeatedly, it is expressed in exponential form. For a number ana^n, aa is called the base and nn is the exponent or power.

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A Pinch of History: Historically, mathematicians like Archimedes used powers of 10 to represent massive quantities, such as the number of grains of sand in the universe (The Sand Reckoner). This led to the modern scientific notation.

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Powers with Negative Exponents: For any non-zero integer aa, a−m=1ama^{-m} = \frac{1}{a^m}, where mm is a positive integer. a−ma^{-m} is the multiplicative inverse of ama^m.

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Laws of Exponents: These rules allow for the simplification of expressions involving powers with the same base or the same exponent.

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Standard Form (Scientific Notation): A way to express very large or very small numbers as the product of a number between 1.01.0 (inclusive) and 10.010.0 and a power of 1010. Formula: k×10nk \times 10^n.

📐Formulae

am×an=am+na^m \times a^n = a^{m+n}

aman=am−n\frac{a^m}{a^n} = a^{m-n}

(am)n=amn(a^m)^n = a^{mn}

am×bm=(ab)ma^m \times b^m = (ab)^m

ambm=(ab)m\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m

a0=1a^0 = 1

a−n=1ana^{-n} = \frac{1}{a^n}

💡Examples

Problem 1:

Evaluate the value of (13)−2\left( \frac{1}{3} \right)^{-2}.

Solution:

(13)−2=32=9\left( \frac{1}{3} \right)^{-2} = 3^2 = 9

Explanation:

Using the law a−n=1ana^{-n} = \frac{1}{a^n}, the reciprocal of the base 13\frac{1}{3} is 33, which changes the exponent from −2-2 to 22.

Problem 2:

Simplify 25÷282^5 \div 2^8 and express the result with a positive exponent.

Solution:

25−8=2−3=1232^{5-8} = 2^{-3} = \frac{1}{2^3}

Explanation:

Applying the quotient law aman=am−n\frac{a^m}{a^n} = a^{m-n}. The subtraction of exponents is as follows: 5−8−3\begin{array}{r} 5 \\ - 8 \\ \hline -3 \end{array} This gives 2−32^{-3}, which is written as 123\frac{1}{2^3} to maintain a positive exponent.

Problem 3:

Express the number 0.0000005640.000000564 in standard form.

Solution:

5.64×10−75.64 \times 10^{-7}

Explanation:

To move the decimal point to the right of the first non-zero digit (55), we shift it 77 places to the right. Since we are moving the decimal to the right (representing a small number), the power of 1010 is negative.