Questions: Power Play

Q1. What is exponential notation? Explain base and exponent with an example.

Answer (English): Exponential notation writes repeated multiplication in a short form. In n^a, the base n is the number being multiplied and the exponent a counts how many copies are used. For example, 5 x 5 x 5 x 5 = 5^4 = 625, where 5 is the base and 4 is the exponent. The expression n^a is read as "n raised to the power a".

Hinglish: Exponential notation baar-baar multiplication ko chhote roop mein likhta hai. n^a mein base n woh number hai jo multiply hota hai, aur exponent a batata hai kitni baar. Jaise 5 x 5 x 5 x 5 = 5^4 = 625, jahan 5 base hai aur 4 exponent. Ise "n raised to the power a" padhte hain.

Q2. State the laws of exponents for multiplication, division and powers of powers.

Answer (English): For the same base, multiplying powers adds the exponents: n^a x n^b = n^{a+b}. Dividing subtracts them: n^a / n^b = n^{a-b}, with n not zero. Raising a power to a power multiplies the exponents: (n^a)^b = n^{ab}. When a product is raised to a power, the exponent applies to each factor: n^a x m^a = (nm)^a.

Hinglish: Same base par multiplication mein exponents add hote hain: n^a x n^b = n^{a+b}. Division mein subtract: n^a / n^b = n^{a-b}, jahan n zero nahi. Power ka power lene par exponents multiply: (n^a)^b = n^{ab}. Product par power ho to har factor par lagta hai: n^a x m^a = (nm)^a.

Q3. Why is any non-zero number to the power zero equal to one?

Answer (English): Use the division law: n^a / n^a = n^{a-a} = n^0. But a non-zero number divided by itself is 1. Therefore n^0 = 1. This is a definition made so that the laws of exponents stay consistent; it is not obtained by multiplying zero factors.

Hinglish: Division law se n^a / n^a = n^{a-a} = n^0. Par koi bhi non-zero number khud se divide karein to 1 aata hai. Isliye n^0 = 1. Ye definition isliye banayi gayi ki exponent laws consistent rahein; ye zero factors multiply karne se nahi aata.

Q4. What does a negative exponent mean?

Answer (English): A negative exponent shows the reciprocal of a power. Since halving corresponds to subtracting one from the exponent, 2^{-1} is one half, 2^{-2} is one quarter, and in general n^{-a} = 1/n^a. For example, 10^{-3} = 1/1000 and 4^{-2} = 1/16. Negative exponents do not make the number negative.

Hinglish: Negative exponent power ka reciprocal dikhata hai. Aadha karna exponent se ek ghatane jaisa hai, isliye 2^{-1} = 1/2, 2^{-2} = 1/4, aur n^{-a} = 1/n^a. Jaise 10^{-3} = 1/1000 aur 4^{-2} = 1/16. Negative exponent se number negative nahi hota.

Q5. Explain the difference between linear growth and exponential growth.

Answer (English): Linear growth adds the same amount at each step, while exponential growth multiplies by the same factor. A ladder with 20 cm between steps is linear: the distance grows by addition. Folding paper doubles the thickness each time: the distance grows by multiplication. After many steps exponential growth becomes far faster, which is why 46 folds can cross the distance that needs about 192 crore ladder steps.

Hinglish: Linear growth mein har step par same amount add hota hai, jabki exponential growth mein same factor se multiply hota hai. 20 cm gap wali ladder linear hai: distance addition se badhti hai. Paper fold karne par thickness har baar double hoti hai: growth multiplication se. Kai steps ke baad exponential growth bahut tez ho jaati hai; isliye 46 folds utni doori paar kar lete hain jiske liye 192 crore ladder steps chahiye.

Q6. What is scientific notation, and why is it useful?

Answer (English): Scientific notation writes a number as x x 10^y, where x is at least 1 and less than 10 and y is an integer. For example, 59,853 = 5.9853 x 10^4 and 0.00045 = 4.5 x 10^{-4}. It is useful for very large or very small numbers because the exponent shows the size at a glance and the coefficient shows the significant digits.

Hinglish: Scientific notation mein number x x 10^y likha jaata hai, jahan x kam se kam 1 aur 10 se kam ho, aur y integer. Jaise 59,853 = 5.9853 x 10^4 aur 0.00045 = 4.5 x 10^{-4}. Bahut bade ya chhote numbers ke liye ye useful hai kyunki exponent size batata hai aur coefficient significant digits dikhata hai.

Q7. How do you count the number of possible lock codes?

Answer (English): Each position of the code is an independent choice, so the number of codes is the product of the choices for the positions. A 5-digit code using 0 to 9 has 10 choices per position, giving 10^5 = 1,00,000 codes. A 5-character code that allows digits and letters has 36 choices per position, giving 36^5 codes.

Hinglish: Code ki har position ek independent choice hai, isliye total codes positions ke choices ka product hota hai. 0 se 9 tak ke digits wale 5-digit code mein har position par 10 choices hain, isliye 10^5 = 1,00,000 codes. Agar 5 characters mein digits aur letters dono allowed hain to har position par 36 choices, isliye 36^5 codes.

Q8. Why do the powers of 10 matter in place value?

Answer (English): Place value is built on powers of 10. The number 47561 is 4 x 10^4 plus 7 x 10^3 plus 5 x 10^2 plus 6 x 10^1 plus 1 x 10^0. After the decimal point the exponents continue as -1, -2 and so on. This is why multiplying or dividing by powers of 10 shifts the decimal point instead of changing the digits.

Hinglish: Place value powers of 10 par based hai. 47561 = 4 x 10^4 + 7 x 10^3 + 5 x 10^2 + 6 x 10^1 + 1 x 10^0. Decimal point ke baad exponents -1, -2 aage chalte hain. Isi wajah se 10 ki powers se multiply ya divide karne par decimal point shift hota hai, digits nahi badalte.

Q9. How do we estimate quantities like the number of steps to the Moon?

Answer (English): Make an instinctive guess first, then build a model with reasonable assumptions and calculate. For the Moon, assume a step length of 20 cm and a distance of about 3,84,400 km. Converting the distance to centimetres and dividing by 20 gives about 1,92,20,00,000 steps. Check whether the guess and the calculation are in the same range; with practice the guesses improve.

Hinglish: Pehle instinctive guess karo, phir reasonable assumptions se model banao aur calculate karo. Moon ke liye step 20 cm aur distance kareeb 3,84,400 km maano. Distance ko centimetres mein badal kar 20 se divide karo, kareeb 1,92,20,00,000 steps milte hain. Guess aur calculation ka range compare karo; practice se guesses behtar hoti hain.

Q10. How can a number that is both a square and a cube be described?

Answer (English): Such numbers are the sixth powers. A square has all prime exponents even, and a cube has all prime exponents divisible by 3; both conditions hold exactly when every exponent is a multiple of 6. So the numbers are n^6, for example 1^6 = 1, 2^6 = 64 and 3^6 = 729. The first example, 64, comes from 2^6 = 8^2 = 4^3.

Hinglish: Aise numbers sixth powers hote hain. Square mein sab prime exponents even hote hain, cube mein sab exponents 3 ke multiples; dono tabhi hote hain jab har exponent 6 ka multiple ho. Isliye numbers n^6 hain, jaise 1^6 = 1, 2^6 = 64, 3^6 = 729. Pehla example 64, 2^6 = 8^2 = 4^3 se aata hai.

The quiz below checks exponents, laws, scientific notation and growth. Review anything that feels shaky.