Assertion (A): The number 0.0000000000085 can be written as 8.5 × 10^-12 in scientific notation. Reason (R): To express a number less than 1 in scientific notation, the decimal point is shifted to the right, and the exponent of 10 is negative, equal to the number of places shifted.

A and R both correct, R explains A

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This quiz covers all key topics from Chapter 2, including exponential notation, understanding base and exponent, various laws of exponents (multiplication, division, power of a power, negative exponents), and scientific notation.

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Home › CBSE › Class 8 › Maths › POWER PLAY

CBSE · Class 8 · 🧮 Maths · Chapter 2

POWER PLAY

Exponential NotationBase and ExponentLaws of ExponentsNegative ExponentsScientific NotationStandard Form

Chapter 2, 'POWER PLAY', introduces students to the fascinating world of exponents and powers. You will learn about exponential notation, where a number is multiplied by itself multiple times, and understand the base and exponent. The chapter also covers the laws of exponents, including multiplication and division of powers with the same base, power of a power, and negative exponents. Finally, you'll explore scientific notation for expressing very large or very small numbers, which is crucial for various scientific applications. Mastering these concepts is fundamental for higher-level mathematics.

Introduction to Exponents

Exponents ya Powers, repeated multiplication ko short form mein likhne ka tareeka hai. Isse bade numbers ko handle karna easy ho jaata hai.

Base aur Exponent:
Ek expression jaise \(a^n\) mein, 'a' ko base kehte hain.
'n' ko exponent ya power kehte hain.
Iska matlab hai 'a' ko 'n' times khud se multiply karna: \(a^n = a \times a \times a \times ... \times a\) (n times).

Examples:
\(2^3 = 2 \times 2 \times 2 = 8\)
\(5^4 = 5 \times 5 \times 5 \times 5 = 625\)
\((-3)^2 = (-3) \times (-3) = 9\)
\((-2)^3 = (-2) \times (-2) \times (-2) = -8\)

Important Points:
Agar base negative ho aur exponent even ho, toh result positive hoga. Example: \((-4)^2 = 16\).
Agar base negative ho aur exponent odd ho, toh result negative hoga. Example: \((-4)^3 = -64\).
\(a^1 = a\) (Kisi bhi number ki power 1 wahi number hota hai).
\(a^0 = 1\) (Kisi bhi non-zero number ki power 0 hamesha 1 hoti hai). Yeh bahut important rule hai!

Reading Exponents:
\(a^2\) ko "a squared" ya "a to the power 2" padhte hain.
\(a^3\) ko "a cubed" ya "a to the power 3" padhte hain.
\(a^n\) ko "a to the power n" padhte hain.

Prime Factorization Method:
Bade numbers ko exponential form mein express karne ke liye prime factorization use karte hain.
Example: 64 ko exponential form mein likho.
\(64 = 2 \times 32 = 2 \times 2 \times 16 = 2 \times 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6\)

Comparing Exponential Numbers:
Agar bases same hain, toh jiska exponent bada hoga, woh number bada hoga. Example: \(2^5 > 2^3\) (32 > 8).
Agar exponents same hain, toh jiska base bada hoga, woh number bada hoga. Example: \(5^2 > 3^2\) (25 > 9).

Common Mistakes:
\(a^n\) ko \(a \times n\) samajhna. Example: \(2^3 \neq 2 \times 3\).
\((-a)^n\) aur \(-a^n\) mein difference na samajhna. \((-2)^4 = 16\) but \(-2^4 = -(2^4) = -16\).

📖Definition

Exponent (ya Power): Ek number jo batata hai ki base ko kitni baar khud se multiply karna hai.

❗Important

Kisi bhi non-zero number ki power 0 hamesha 1 hoti hai. \(a^0 = 1\) (jahan \(a \neq 0\)).

Laws of Exponents

Exponents ke kuch rules hote hain jo calculations ko simplify karte hain. Inhe Laws of Exponents kehte hain. Yeh rules positive aur negative dono exponents par apply hote hain.

Law 1: Multiplication with Same Base
Jab bases same hote hain, toh powers add ho jaati hain.
Formula: \(a^m \times a^n = a^{m+n}\)
Example: \(2^3 \times 2^4 = 2^{3+4} = 2^7 = 128\)
Example: \((-3)^2 \times (-3)^3 = (-3)^{2+3} = (-3)^5 = -243\)

Law 2: Division with Same Base
Jab bases same hote hain, toh powers subtract ho jaati hain.
Formula: \(a^m \div a^n = a^{m-n}\) (jahan \(a \neq 0\))
Example: \(5^6 \div 5^2 = 5^{6-2} = 5^4 = 625\)
Example: \(x^7 \div x^3 = x^{7-3} = x^4\)

Law 3: Power of a Power
Jab ek power ki power hoti hai, toh powers multiply ho jaati hain.
Formula: \((a^m)^n = a^{m \times n}\)
Example: \((3^2)^3 = 3^{2 \times 3} = 3^6 = 729\)
Example: \(( ( -2 )^3 )^2 = ( -2 )^{3 \times 2} = ( -2 )^6 = 64\)

Law 4: Power of a Product
Jab product ki power hoti hai, toh har factor ki power ho jaati hai.
Formula: \((ab)^m = a^m b^m\)
Example: \((2 \times 3)^4 = 2^4 \times 3^4 = 16 \times 81 = 1296\)
Example: \((xy)^5 = x^5 y^5\)

Law 5: Power of a Quotient
Jab quotient ki power hoti hai, toh numerator aur denominator dono ki power ho jaati hai.
Formula: \((a/b)^m = a^m / b^m\) (jahan \(b \neq 0\))
Example: \((2/5)^3 = 2^3 / 5^3 = 8 / 125\)
Example: \((p/q)^2 = p^2 / q^2\)

Law 6: Zero Exponent
Kisi bhi non-zero base ki power 0 hamesha 1 hoti hai.
Formula: \(a^0 = 1\) (jahan \(a \neq 0\))
Example: \(7^0 = 1\)
Example: \((-100)^0 = 1\)
Example: \((x+y)^0 = 1\) (jahan \(x+y \neq 0\))

Important Note: Yeh laws tabhi apply hote hain jab bases ya exponents same hon, jaisa ki har law mein specify kiya gaya hai. Careful rehna calculations karte waqt!

🧮Formula

Laws of Exponents Summary:

\(a^m \times a^n = a^{m+n}\)
\(a^m \div a^n = a^{m-n}\)
\((a^m)^n = a^{mn}\)
\((ab)^m = a^m b^m\)
\((a/b)^m = a^m / b^m\)
\(a^0 = 1\) (for \(a \neq 0\))

🚧Misconception

Aksar students \(a^m + a^n\) ko \(a^{m+n}\) samajh lete hain. Yeh galat hai! Addition mein powers add nahi hoti. Example: \(2^2 + 2^3 = 4 + 8 = 12\), but \(2^{2+3} = 2^5 = 32\).

Numbers with Negative Exponents

Ab tak humne positive aur zero exponents dekhe. Ab dekhte hain negative exponents ka kya matlab hota hai.

Definition of Negative Exponent:
Agar kisi non-zero number 'a' ki power negative 'n' hai, toh uska matlab hai 1 ko \(a^n\) se divide karna.
Formula: \(a^{-n} = 1/a^n\) (jahan \(a \neq 0\))
Example: \(2^{-3} = 1/2^3 = 1/8\)
Example: \(5^{-2} = 1/5^2 = 1/25\)

Fractional Base with Negative Exponent:
Agar base fraction mein hai aur power negative hai, toh fraction ko reciprocal kar dete hain aur power positive ho jaati hai.
Formula: \((a/b)^{-n} = (b/a)^n\) (jahan \(a, b \neq 0\))
Example: \((2/3)^{-2} = (3/2)^2 = 3^2 / 2^2 = 9/4\)
Example: \((1/5)^{-3} = (5/1)^3 = 5^3 = 125\)

Applying Laws of Exponents with Negative Exponents:
Laws of exponents negative exponents ke liye bhi valid hain.
Example (Law 1): \(2^{-3} \times 2^5 = 2^{-3+5} = 2^2 = 4\)
Example (Law 2): \(3^2 \div 3^{-3} = 3^{2 - (-3)} = 3^{2+3} = 3^5 = 243\)
Example (Law 3): \(( ( 4 )^{-2} )^3 = 4^{-2 \times 3} = 4^{-6} = 1/4^6 = 1/4096\)

Simplifying Expressions:
Jab bhi koi expression simplify karna ho, koshish karo ki final answer mein positive exponents hi hon.
Example: Simplify \((2^{-1} \times 4^{-1}) \div 2^{-2}\)
\((1/2 \times 1/4) \div (1/2^2)\)
\((1/8) \div (1/4)\)
\(1/8 \times 4/1 = 4/8 = 1/2\)

Reciprocal Relationship:
\(a^{-n}\) aur \(a^n\) ek doosre ke reciprocal hote hain. Matlab \(a^{-n} = 1/a^n\) aur \(a^n = 1/a^{-n}\).
Example: \(2^{-3}\) ka reciprocal \(2^3\) hai.

Key Takeaway: Negative exponent ka matlab number negative nahi hota, balki uska reciprocal hota hai. Yeh confusion avoid karna!

🧮Formula

Negative Exponent Rule: \(a^{-n} = 1/a^n\) (jahan \(a \neq 0\)) \((a/b)^{-n} = (b/a)^n\) (jahan \(a, b \neq 0\))

💡Tip

Final answer mein hamesha positive exponents rakho, jab tak question mein specifically negative exponents mein answer dene ko na kaha gaya ho. Yeh ek common instruction hoti hai exams mein.

Expressing Large and Small Numbers in Standard Form

Bahut bade ya bahut chhote numbers ko likhne aur unke saath calculations karne mein mushkil hoti hai. Isliye unhe Standard Form ya Scientific Notation mein express karte hain.

Standard Form Definition:
Ek number ko standard form mein tab kaha jaata hai jab use \(k \times 10^n\) ke roop mein likha jaata hai.
Jahan \(k\) ek decimal number hota hai jo \(1 \le k < 10\) ki condition satisfy karta hai (matlab \(k\) 1 ya 1 se bada aur 10 se chhota hota hai).
Aur \(n\) ek integer hota hai (positive ya negative).

Expressing Large Numbers in Standard Form:
Decimal point ko left side move karte hain jab tak first non-zero digit ke baad na aa jaye.
Jitne places decimal move kiya, woh \(10\) ki positive power ban jaati hai.
Example: 3,450,000,000
Decimal ko 9 places left move kiya: 3.45
So, \(3.45 \times 10^9\)
Example: 89,000
Decimal ko 4 places left move kiya: 8.9
So, \(8.9 \times 10^4\)

Expressing Small Numbers in Standard Form:
Decimal point ko right side move karte hain jab tak first non-zero digit ke baad na aa jaye.
Jitne places decimal move kiya, woh \(10\) ki negative power ban jaati hai.
Example: 0.0000000078
Decimal ko 9 places right move kiya: 7.8
So, \(7.8 \times 10^{-9}\)
Example: 0.000123
Decimal ko 4 places right move kiya: 1.23
So, \(1.23 \times 10^{-4}\)

Converting Standard Form to Usual Form:
Agar power positive hai, toh decimal ko right move karte hain.
Agar power negative hai, toh decimal ko left move karte hain.
Example: \(6.02 \times 10^5\)
Decimal ko 5 places right move kiya: 602000
Example: \(3.1 \times 10^{-3}\)
Decimal ko 3 places left move kiya: 0.0031

Uses of Standard Form:
Astronomy mein (stars aur planets ke distances).
Physics mein (atoms aur electrons ke sizes).
Chemistry mein (number of molecules).
Computer science mein (data storage).

Remember: Standard form mein hamesha ek hi non-zero digit decimal point ke left mein hona chahiye. Yeh rule mat bhoolna!

📖Definition

Standard Form (Scientific Notation): Ek number ko \(k \times 10^n\) ke roop mein likhna, jahan \(1 \le k < 10\) aur \(n\) ek integer hai.

⭐Remember

Decimal ko left move karne par power positive hoti hai. Decimal ko right move karne par power negative hoti hai.

Easy (15)

In the expression 7^5, what is the base?
- 5
- 7 (correct)
- 75
- 1
Why: The base is the number being multiplied, which is 7.
What is the value of 2^3?
- 6
- 8 (correct)
- 9
- 5
Why: 2^3 means 2 multiplied by itself 3 times, which is 2 × 2 × 2 = 8.
Express 4 × 4 × 4 in exponential form.
- 3^4
- 4^3 (correct)
- 4 × 3
- 12
Why: The base is 4 and it is multiplied 3 times, so it's 4^3.
Any non-zero number raised to the power of zero is equal to:
- 0
- 1 (correct)
- The number itself
- Undefined
Why: A fundamental rule of exponents states that any non-zero number raised to the power of zero is 1.
What is the value of 10^-2?
- 100
- -100
- 0.01 (correct)
- -0.01
Why: 10^-2 is 1 divided by 10^2, which is 1/100 or 0.01.
Which of the following is the standard form of 3.081 × 10^8?
- 30810000
- 308100000 (correct)
- 3081000000
- 3081000
Why: Move the decimal 8 places to the right to convert from scientific to standard form.
The expression (a^m)^n is equivalent to:
- a^(m+n)
- a^(m-n)
- a^(m*n) (correct)
- a^(m/n)
Why: According to the power of a power rule, exponents are multiplied.
What is the result of 5^2 × 5^3?
- 5^6
- 25^6
- 5^5 (correct)
- 25^5
Why: When multiplying powers with the same base, add the exponents: 2 + 3 = 5.
The value of (-1)^even number is always 1.
Why: When -1 is multiplied an even number of times, the negative signs cancel out in pairs, resulting in a positive 1.
The scientific notation for 0.000005 is 5 × 10^-6.
Why: To get 5 from 0.000005, the decimal point moves 6 places to the right, hence 10^-6.
Exponential growth refers to multiplicative growth.
Why: Exponential growth means a quantity increases by a constant factor over equal intervals, which is multiplicative.
The value of (2/3)^-2 is 9/4.
Why: A negative exponent means taking the reciprocal of the base and raising it to the positive exponent.
In the expression 5^4, the number 4 is called the _________.
Why: The number indicating how many times the base is multiplied is the exponent.
The standard form of scientific notation for any number is x × 10^y, where x ≥ 1 and x < _________.
Why: In standard scientific notation, the number 'x' must be between 1 (inclusive) and 10 (exclusive).
The initial thickness of a paper folded once becomes its original thickness multiplied by _________.
Why: Each fold doubles the thickness of the paper.

Medium (15)

Simplify: (3^2)^3 × 3^4
- 3^9
- 3^10 (correct)
- 3^12
- 3^24
Why: First, apply (a^m)^n = a^(m*n), then a^m × a^n = a^(m+n).
If 2^x = 1/32, what is the value of x?
- 5
- -5 (correct)
- 1/5
- -1/5
Why: Since 32 = 2^5, 1/32 = 2^-5, so x = -5.
The value of (5^0 + 4^0 + 3^0) is:
- 0
- 1
- 3 (correct)
- 12
Why: Any non-zero number raised to the power of zero is 1. So, 1 + 1 + 1 = 3.
Simplify: (2^5 ÷ 2^8)^5 × 2^-5
- 2^10
- 2^-10
- 2^-20 (correct)
- 2^-30
Why: Simplify inside the parenthesis first: 2^(5-8) = 2^-3. Then (2^-3)^5 = 2^-15. Finally, 2^-15 × 2^-5 = 2^(-15-5) = 2^-20.
The value of (1/3)^-2 is:
- 1/9
- -9
- 9 (correct)
- -1/9
Why: A negative exponent means taking the reciprocal of the base and raising it to the positive exponent: (3/1)^2 = 9.
Match the exponential expressions with their simplified forms.
Why: Apply the laws of exponents: add exponents for multiplication, multiply for power of a power, subtract for division, and any non-zero number to power zero is 1.
Match the numbers with their scientific notation.
Why: Adjust the decimal point to have one non-zero digit before it, and count the shifts for the exponent.
Match the expressions with their equivalent forms.
Why: These are the fundamental laws of exponents.
What is the value of (1/2)^-3?
Why: Flip the base and make the exponent positive: (2/1)^3 = 2^3 = 8.
Express 125 as a power of 5.
Why: 125 = 5 × 5 × 5, which is 5 raised to the power of 3.
Arrange the steps to simplify (2^3 × 2^-5)^-2 in correct order.
Why: Follow the order of operations and laws of exponents.
Order the following numbers from smallest to largest: 3^2, 2^3, 4^1, 1^5.
Why: Calculate the value of each expression and then order them.
Which diagram correctly represents 3^4?
- Diagram A
- Diagram B (correct)
- Diagram C
- Diagram D
Why: 3^4 means 3 multiplied by itself 4 times.
The diagram shows the thickness of a paper after several folds. If the initial thickness was 't', which diagram correctly represents the thickness after 3 folds?
- Diagram P
- Diagram Q (correct)
- Diagram R
- Diagram S
Why: After 3 folds, the thickness becomes t × 2^3 = 8t.
Which diagram illustrates the concept of a^-n = 1/a^n?
- Diagram X
- Diagram Y (correct)
- Diagram Z
- None of the above
Why: Diagram Y shows a reciprocal relationship with a positive exponent.

Hard (10)

What is the value of ( (1/3)^-2 - (1/2)^-3 ) ÷ (1/4)^-2?
Why: Simplify each term using negative exponent rule, then perform subtraction and division.
Find the value of m if (-3)^(m+1) × (-3)^5 = (-3)^7.
Why: Using a^m × a^n = a^(m+n), equate the exponents: m+1+5 = 7.
If (5/3)^-5 × (5/3)^11 = (5/3)^(8x), find the value of x.
Why: Combine exponents on the left: (5/3)^(-5+11) = (5/3)^6. Then equate 6 = 8x.
The distance between Earth and Moon is approximately 3.84 × 10^8 meters. Express this distance in kilometers in standard form.
Why: 1 km = 1000 m = 10^3 m. So divide the distance in meters by 10^3.
Assertion (A): The value of (2^3)^2 is 64. Reason (R): When a power is raised to another power, the exponents are added.
Why: Assertion is true (2^6 = 64), but the reason incorrectly states exponents are added for power of a power; they are multiplied.
Assertion (A): The number 0.0000000000085 can be written as 8.5 × 10^-12 in scientific notation. Reason (R): To express a number less than 1 in scientific notation, the decimal point is shifted to the right, and the exponent of 10 is negative, equal to the number of places shifted.
Why: Both assertion and reason are correct, and the reason accurately explains the process of converting small numbers to scientific notation.
A bacteria culture initially contains 1000 bacteria. If the number of bacteria doubles every hour, how many bacteria will there be after 5 hours? Express your answer in scientific notation.
- 3.2 × 10^4 (correct)
- 1.6 × 10^4
- 1.0 × 10^3
- 3.2 × 10^3
Why: Initial bacteria × 2^(number of hours) = 1000 × 2^5 = 1000 × 32 = 32000 = 3.2 × 10^4.
The speed of light is approximately 3 × 10^8 meters per second. A star is 9.46 × 10^15 meters away from Earth. How long (in seconds) would it take for light from that star to reach Earth? Express your answer in scientific notation.
- 3.153 × 10^7 (correct)
- 3.153 × 10^8
- 2.838 × 10^23
- 3.153 × 10^-7
Why: Time = Distance / Speed. Divide (9.46 × 10^15) by (3 × 10^8).
The diagram shows a cube with side length 'x'. If its volume is 216 cubic units, which expression correctly represents the side length 'x'?
- x = 216^2
- x = 216^3
- x = 216^(1/3) (correct)
- x = 216^(-1/3)
Why: Volume of a cube is side^3. So, side = (Volume)^(1/3).
A diagram shows a sequence of squares. The first square has side length 1 unit. The second square has side length 2 units. The third square has side length 4 units. If this pattern continues, which expression represents the side length of the nth square?
- 2n
- 2^(n-1) (correct)
- n^2
- 2^n
Why: The side lengths are 1, 2, 4, which are 2^0, 2^1, 2^2. So for the nth square, it's 2^(n-1).