2.7 Going Deeper: Enrichment & Exam Preparation

Where Powers Show Up in Real Life

Math in the Real World

  • Sizing the universe and the atom. Scientific notation is how scientists tame numbers too big or too small to write out. The Sun is about 1.496\times 10^{11} m away; light travels 3\times 10^8 m every second; a hydrogen atom is roughly 1\times 10^{-10} m across. The exponent alone tells you the scale at a glance.
  • Digital storage in powers of 2. Computer memory is counted in powers of two: 1\text{ KB} = 2^{10} = 1024 bytes, 1\text{ MB} = 2^{20} bytes, and 1\text{ GB} = 2^{30} bytes, over a billion. That is why a “256 GB” phone and a “2^8 GB” phone are the very same thing.
  • Doubling and growth. Money at compound interest, a rumour passing person to person, bacteria splitting in a dish, all grow multiplicatively, like the folded foil. A quantity that doubles every step reaches 2^{10} = 1024 times its start in just ten steps.
  • Earthquakes and sound. The Richter scale and the decibel scale are logarithmic: each step up multiplies by a fixed power of ten, so a magnitude-7 quake releases about 10^2 = 100 times the energy of a magnitude-5 one.
  • A bridge to later chapters. The very same index laws will reappear when you handle algebraic expressions (Chapter 3) and shrink huge direct- and inverse-proportion quantities into scientific notation for comparison.

Quick Reference: Laws of Exponents & Key Powers

Knowing these by heart turns slow calculations into instant recall, a real advantage under exam time pressure. (Throughout, n \neq 0.)

Law Rule Example
Product (same base) n^a \times n^b = n^{a+b} 2^3\times 2^4 = 2^7 = 128
Quotient (same base) n^a \div n^b = n^{a-b} 5^6 \div 5^2 = 5^4 = 625
Power of a power (n^a)^b = n^{ab} (3^2)^3 = 3^6 = 729
Same exponent (product) m^a \times n^a = (mn)^a 4^3\times 2^3 = 8^3 = 512
Same exponent (quotient) \dfrac{n^a}{m^a} = \left(\dfrac{n}{m}\right)^a \dfrac{12^4}{4^4} = 3^4 = 81
Zero index n^0 = 1 7^0 = 1
Negative index n^{-a} = \dfrac{1}{n^a} 2^{-3} = \dfrac{1}{8}
n 2^n n 2^n Power of 10 Value
1 2 7 128 10^0 1
2 4 8 256 10^1 10
3 8 9 512 10^2 100
4 16 10 1024 10^3 1000
5 32 11 2048 10^4 10 000
6 64 12 4096 10^6 10 lakh

Use the converter to turn any number into scientific notation and back, and watch the decimal point slide:

Scientific-notation converter

Memory Tricks & One-Page Revision

Quick Revision Card

  • Same base, multiply → ADD the indices. Same base, divide → SUBTRACT. Power of a power → MULTIPLY.
  • Zero index: anything (non-zero) to the power 0 is 1, so 5^0 = 1, but in 5\times x^0 only the x becomes 1, leaving 5.
  • Negative index = flip it: n^{-a} = \tfrac{1}{n^a}. The sign of the index never reaches the value; 2^{-3} = \tfrac18 is still positive.
  • Sliding the decimal: \times 10^a moves the point a places right; \times 10^{-a} moves it a places left.
  • Scientific notation: x\times 10^y with 1 \le x < 10. The exponent carries the size; one step up means ten times bigger.
  • Negative base: odd power keeps the minus, even power kills it. (-2)^5 = -32 but (-2)^4 = +16.
  • Powers of 2 to know cold: 2^{10}=1024, so a thousand-ish, a million-ish, a billion-ish are 2^{10},2^{20},2^{30}, exactly KB, MB, GB.

Spot the Mistake

Common Exam Mistakes

  • Multiplying the indices when you should add. 2^3 \times 2^4 = 2^{7} (add), not 2^{12}.
  • Thinking n^0 = 0. Any non-zero base to the power 0 is 1, not 0.
  • Believing a negative index makes the answer negative. 3^{-2} = \tfrac19, a positive fraction, not -9.
  • Letting the coefficient drift outside 1\le x<10. Writing 48.6\times 10^5 is not proper scientific notation; it should be 4.86\times 10^6.
  • Raising the wrong factor. In 5\times 3^4, only the 3 is raised; the answer is 5\times 81 = 405, not 15^4.
  • Mishandling a negative base’s sign. (-2)^4 = 16 (even power, positive), while -2^4 = -16 (here the power binds before the minus sign).

Test Your Reflexes

A fast, friendly drill on the laws of exponents: simplify each expression to a single power and type the index. Build a streak!

Index reflex
Simplify to a single power of n and type the resulting index.
n³ × n⁴ = n?
Score 0 · Streak 0

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. Simplify: \dfrac{2^5 \times 2^3}{2^4}.
  2. Evaluate 7^0 + 3^0.
  3. Write 3^{-2} as a fraction.

Short answer (2 marks each)

  1. Simplify \dfrac{2^3 \times 2^2}{2^4} and write the result as a single power of 2 and as a whole number.
  2. Express 0.000045 in scientific notation, and write 5.6\times 10^7 as an ordinary number.

Long answer (3 marks each)

  1. Simplify \dfrac{5^{-3}\times 5^{2}}{5^{-4}}, giving the answer first as a single power of 5 and then as a whole number.
  2. A single bacterium splits into two every hour. Write, in exponential form, the number present after 12 hours, give the actual number, and express it in scientific notation.

HOTS (Higher Order Thinking)

  1. Which is greater, 2^{10} or 10^3? Justify your answer.
  2. Find the units digit of 7^{100}. (Hint: list the units digits of 7^1, 7^2, 7^3, 7^4 and look for the repeating cycle.)

Assertion–Reason (Choose: (a) both true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.)

  1. Assertion (A): (2^3)^2 = 2^6 = 64.   Reason (R): Raising a power to a power multiplies the two indices.
  1. Add the indices on top, subtract the one below: 2^{5+3-4} = 2^{4} = \mathbf{16}.
  2. Any non-zero base to the power 0 is 1, so 7^0 + 3^0 = 1 + 1 = \mathbf{2}.
  3. 3^{-2} = \dfrac{1}{3^2} = \mathbf{\dfrac{1}{9}}.
  4. \dfrac{2^3 \times 2^2}{2^4} = 2^{3+2-4} = 2^{1} = \mathbf{2}.
  5. 0.000045 = \mathbf{4.5\times 10^{-5}} (the point moves 5 places right to rebuild it); and 5.6\times 10^7 = \mathbf{5{,}60{,}00{,}000}.
  6. \dfrac{5^{-3}\times 5^{2}}{5^{-4}} = 5^{-3+2-(-4)} = 5^{-3+2+4} = 5^{3} = \mathbf{125}.
  7. Each hour doubles the count: after 12 hours there are 2^{12} bacteria = \mathbf{4096}, which in scientific notation is \mathbf{4.096\times 10^{3}}.
  8. 2^{10} = 1024 while 10^3 = 1000, so \mathbf{2^{10}} is greater (by 24).
  9. Units digits of 7^1, 7^2, 7^3, 7^4 are 7, 9, 3, 1, then the cycle of length 4 repeats. Since 100 = 4\times 25 is a multiple of 4, the units digit is the last in the cycle: \mathbf{1}.
  10. (a), Both statements are true and R is the correct explanation: (2^3)^2 = 2^{3\times 2} = 2^6 = 64 exactly because the power-of-a-power law multiplies the indices.

Connections to Other Chapters

How This Chapter Links Forward

  • Chapter 1 (A Square and a Cube): n^2 and n^3 were simply powers with index 2 and 3, this chapter generalises squaring and cubing to any index.
  • Chapter 3 (Algebraic Expressions & Identities): the same product, quotient and power laws govern algebraic terms such as x^4\cdot x^3 = x^7 and (a^2)^3 = a^6.
  • Chapters on Proportion & Mensuration: scientific notation lets you compare and combine very large or very small measured quantities cleanly.

Glossary

Key Terms

  • Base: the number being multiplied repeatedly; the n in n^a.
  • Exponent / index / power: how many times the base is used as a factor; the a in n^a.
  • Power: the whole expression n^a (and informally the exponent itself).
  • Laws of exponents: the rules for combining powers, product, quotient, power-of-a-power, and same-exponent.
  • Zero exponent: the convention n^0 = 1 for any non-zero n.
  • Negative exponent: n^{-a} = \tfrac{1}{n^a}, signalling a reciprocal.
  • Scientific (standard) notation: writing a number as x\times 10^y with 1\le x<10 and integer y.
  • Coefficient: the value x in x\times 10^y, holding the leading digits.
  • Exponential growth: repeated multiplication by a fixed factor; contrasts with additive linear growth.