Power Play Class 8: Exponents, Laws of Exponents and Scientific Notation

What Power Play is about

Key concepts

Exponential growth: the paper-folding story

NCERT section 2.1 Experiencing the Power Play ...

When a quantity is multiplied by the same number again and again, it grows very quickly. Folding paper doubles the thickness each time:

• after 1 fold: $0.001\times 2$ cm; after 2 folds: $0.001\times 2\times 2$ cm; after $n$ folds: $0.001\times 2^n$ cm.
• After 10 folds the thickness is $0.001\times 1024\approx 1$ cm; after 20 folds about 10.5 m; after 30 folds about 10.7 km.

Any 10 extra folds multiply the thickness by $2^{10}=1024$, so each block of 10 folds makes the stack roughly a thousand times thicker.

Linear growth adds the same amount each step (a ladder rung every 20 cm); exponential growth multiplies by the same amount each step. A ladder to the Moon would need about 192 crore rungs, but folding needs only 46 doublings.

Worked example

Suppose a sheet is 0.002 cm thick. How thick would it be after 15 folds? After how many folds is it first thicker than 1 cm?

Try it first, then show the solutionSolution

1. After $n$ folds the thickness is $0.002\times 2^n$ cm.
2. $2^{15}=32768$, so after 15 folds it is $0.002\times 32768=65.536$ cm.
3. We need $0.002\times 2^n>1$, that is $2^n>500$. $2^8=256$ is too small and $2^9=512$ works.

Answer: About 65.5 cm after 15 folds; it first passes 1 cm after 9 folds.

Base, exponent and powers of negative numbers

NCERT section 2.2 Exponential Notation and Operations

$n^a$ means $n$ multiplied by itself $a$ times. In $5^4=5\times 5\times 5\times 5=625$, 5 is the base and 4 is the exponent (power); $5^4$ is the exponential form of 625, read '5 raised to the power 4'.

• Do not confuse repeated addition with repeated multiplication: $4+4+4=3\times 4=12$, but $4\times 4\times 4=4^3=64$.
• Letters work the same way: $a\times a\times a\times b\times b=a^3{b}^2$.
• Negative bases: an even power of a negative number is positive, an odd power is negative. $(-4{)}^3=-64$, $(-2{)}^4=16$, $(-1{)}^{\text{odd}}=-1$, $(-1{)}^{\text{even}}=1$.
• $0^n=0$ for any counting number $n$, and $1^n=1$.

Prime factors in exponential form: $2700=2\times 2\times 3\times 3\times 3\times 5\times 5=2^2\times 3^3\times 5^2$.

Worked example

Find the value of $(-1{)}^{55}+(-1{)}^{56}$ and of $(-2{)}^5$. Write 2,700 as a product of powers of primes.

Try it first, then show the solutionSolution

1. 55 is odd, so $(-1{)}^{55}=-1$; 56 is even, so $(-1{)}^{56}=1$. Sum $=0$.
2. $(-2{)}^5=(-2)(-2)(-2)(-2)(-2)=-32$ (odd power of a negative number).
3. $2700=27\times 100=3^3\times 2^2\times 5^2$.

Answer: 0; $-32$; $2700=2^2\times 3^3\times 5^2$.

Laws of exponents for multiplication

NCERT section 2.2 Exponential Notation and Operations

Counting the factors gives three rules (for counting numbers $a$, $b$ to begin with):

• Same base, multiply: add the exponents. $n^a\times n^b=n^{a+b}$. For example $3^4\times 3^3=3^7$ (four 3s and three 3s make seven 3s).
• Power of a power: multiply the exponents. $(n^a{)}^b=(n^b{)}^a=n^{ab}$. For example $4^6=(4^3{)}^2=(4^2{)}^3$, and $2^{10}=(2^5{)}^2=(2^2{)}^5$.
• Same exponent, multiply the bases. $m^a\times n^a=(mn{)}^a$; also $\frac{m^a}{n^a}={\left(\frac{m}{n}\right)}^a$. So $2^5\times 5^5=10^5$ and $\frac{10^4}{5^4}=2^4$.

These rules let you work out big powers from smaller ones: $3^7=3^4\times 3^3=81\times 27=2187$.

The Stones that Shine idea: in a tree where every item branches into 3 at each of 7 levels, the count at the bottom is $3^7$.

Worked example

Simplify $2^5\times 5^5$ and $\frac{(3^4{)}^2\times 3^3}{3^9}$.

Try it first, then show the solutionSolution

1. Same exponent: $2^5\times 5^5=(2\times 5{)}^5=10^5=1,00,000$.
2. $(3^4{)}^2=3^8$, and $3^8\times 3^3=3^{11}$.
3. $3^{11}÷3^9=3^2=9$.

Answer: $10^5=1,00,000$; 9.

Counting combinations with powers

NCERT section 2.2 Exponential Notation and Operations

If one choice can be made in $p$ ways and, for each of these, a second choice in $q$ ways, together there are $p\times q$ ways. Repeat this for every choice.

• 4 dresses and 3 caps: $4\times 3=12$ outfits; add 2 pairs of shoes: $4\times 3\times 2=24$.
• A lock with $k$ slots, each showing the digits 0 to 9: $10\times 10\times \cdots =10^k$ codes. A 5-digit lock has $10^5=1,00,000$ codes, from 00000 to 99999.
• Slots with letters A to Z: $26^k$ codes. Slots that may hold a digit or a letter: $36^k$ codes.

Problem-solving tip: if a problem is too big, solve a smaller version first (a 2-digit lock), spot the pattern, then scale up.

Worked example

A bag lock has 3 slots for letters (A to Z) followed by 2 slots for digits. How many codes are possible? How many times as many as a 5-digit lock?

Try it first, then show the solutionSolution

1. Letters: $26\times 26\times 26=26^3=17576$.
2. Digits: $10^2=100$.
3. Total codes $=17576\times 100=17,57,600$.
4. A 5-digit lock has $10^5=1,00,000$ codes; $1757600÷100000=17.576$.

Answer: 17,57,600 codes, about 17.6 times as many as a 5-digit lock.

Division, zero exponent and negative exponents

NCERT section 2.3 The Other Side of Powers

Same base, divide: subtract the exponents. $n^a÷n^b=n^{a-b}$ ($n\ne 0$). Halving a 16-unit line three times: $2^4÷2^3=2^1=2$ units.

Zero exponent. For the rule to keep working when $a=b$: $2^4÷2^4=2^0$, but any non-zero number divided by itself is 1. So $n^0=1$ for every $n\ne 0$.

Negative exponents. Halving 16 units five times gives $2^4÷2^5=2^{-1}$, which is the length $\frac{1}{2}$. In general

• $n^{-a}=\frac{1}{n^a}$ and $n^a=\frac{1}{n^{-a}}$ ($n\ne 0$).
• $10^{-3}=\frac{1}{1000}=0.001$, $7^{-2}=\frac{1}{49}$.

All the laws ($n^a\times n^b=n^{a+b}$, $n^a÷n^b=n^{a-b}$, $(n^a{)}^b=n^{ab}$) still hold when $a$ and $b$ are any integers. Why can the base not be 0? Because $0^a÷0^b$ would mean dividing by 0.

Power lines: on a line of powers of 4 ($\dots ,4^{-2},4^{-1},4^0,4^1,4^2,\dots$), each step right multiplies by 4 and each step left divides by 4. $4^7$ is $4^2=16$ times $4^5$.

Worked example

Simplify $2^{-4}\times 2^7$ and $\frac{5^{-2}\times 5^7}{5^3}$. Write $2^{-3}$ as a fraction and a decimal.

Try it first, then show the solutionSolution

1. $2^{-4}\times 2^7=2^{-4+7}=2^3=8$.
2. $5^{-2}\times 5^7=5^5$, and $5^5÷5^3=5^2=25$.
3. $2^{-3}=\frac{1}{2^3}=\frac{1}{8}=0.125$.

Answer: 8; 25; $2^{-3}=\frac{1}{8}=0.125$.

Powers of 10 and scientific notation

NCERT section 2.4 Powers of 10

Expanded form with powers of 10: $47561=4\times 10^4+7\times 10^3+5\times 10^2+6\times 10^1+1\times 10^0$. Decimals use negative powers: $561.903=5\times 10^2+6\times 10^1+1\times 10^0+9\times 10^{-1}+0\times 10^{-2}+3\times 10^{-3}$.

Scientific notation (standard form): write a number as $x\times 10^y$ where $1\le x<10$ and $y$ is an integer.

• $5900=5.9\times 10^3$; $80,00,000=8\times 10^6$; $3,07,000=3.07\times 10^5$.
• Count how many places the decimal point moves: to the left gives a positive $y$, to the right (for small numbers) a negative $y$: $0.000308=3.08\times 10^{-4}$.

Why it helps: big numbers with many zeros are easy to misread. The exponent tells you the size (how many digits); changing it by 1 changes the number tenfold, while changing the first digit of $x$ changes it much less. The number of digits kept in $x$ shows how precisely the quantity is known: $1.42\times 10^5$ claims more precision than $1.4\times 10^5$.

Comparing: first compare exponents; if equal, compare the coefficients. $1.496\times 10^{11}$ is smaller than $1.4335\times 10^{12}$ because $11<12$.

Worked example

Write 7,04,00,00,000 and 0.000308 in scientific notation. Which is larger, $9.8\times 10^7$ or $1.2\times 10^8$?

Try it first, then show the solutionSolution

1. 7,04,00,00,000 has 10 digits; move the point 9 places left: $7.04\times 10^9$.
2. 0.000308: move the point 4 places right to get 3.08, so $3.08\times 10^{-4}$.
3. Compare exponents first: $10^8>10^7$, so $1.2\times 10^8$ is larger (it is 12 crore; the other is 9.8 crore).

Answer: $7.04\times 10^9$; $3.08\times 10^{-4}$; $1.2\times 10^8$ is larger.

Estimating large quantities and naming big numbers

NCERT section 2.5 Did You Ever Wonder?

Many real questions (how many coins weigh as much as you, how long a pilgrimage walk took) have no exact answer. The book's method:
1. Guess instinctively first (hundreds? lakhs? crores?).
2. Describe the relationships among the quantities, for example number of coins $=$ your mass $÷$ mass of one coin.
3. Make reasonable assumptions for unknowns and compute.
4. Compare with your guess. Different assumptions give different but reasonable answers.

Number names:

• Indian: lakh $=10^5$, crore $=10^7$, arab $=10^9$, kharab $=10^{11}$, neel $=10^{13}$, padma $=10^{15}$, shankh $=10^{17}$ (each is 100 times the previous).
• International: million $=10^6$, billion $=10^9$, trillion $=10^{12}$, quadrillion $=10^{15}$ (each is 1000 times the previous).
• A googol is $10^{100}$.

History: the Buddhist text *Lalitavistara* (first century BCE) names odd powers of ten up to $10^{53}$; Mahaviracharya's *Ganita-sara-sangraha* lists 24 names (up to $10^{23}$), and later texts go far higher.

Time in seconds: $10^5$ s is a little over a day, $10^6$ s about 11.6 days, $10^9$ s about 31.7 years.

Worked example

Suppose a walker takes steps of 25 cm. How many steps are needed to cover 5 km? Compare with the number of times a 0.001 cm sheet must be doubled to be at least 5 km thick.

Try it first, then show the solutionSolution

1. 5 km $=5\times 10^5$ cm. Steps: $5\times 10^5÷25=2\times 10^4=20,000$ (linear growth).
2. Doubling: we need $0.001\times 2^n\ge 5\times 10^5$ cm, so $2^n\ge 5\times 10^8$.
3. $2^{28}\approx 2.68\times 10^8$ is too small; $2^{29}\approx 5.37\times 10^8$ works. So 29 doublings (about 5.37 km).

Answer: 20,000 steps, but only 29 doublings.

Maths in real life: Power Play

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Laws of Exponents, Zero and Negative Powers
• Powers of 10 and Scientific Notation

Power Play formula sheet

Product of powers

$n^a\times n^b=n^{a+b}$

Quotient of powers

$n^a÷n^b=n^{a-b},n\ne 0$

Power of a power

$(n^a{)}^b=(n^b{)}^a=n^{ab}$

Same exponent

$m^a\times n^a=(mn{)}^a,\frac{m^a}{n^a}={\left(\frac{m}{n}\right)}^a$

Zero exponent

$n^0=1,n\ne 0$

Negative exponent

$n^{-a}=\frac{1}{n^a},n^a=\frac{1}{n^{-a}}$

Scientific notation

$x\times 10^y,1\le x<10,y\text{ an integer}$

Combinations

$\text{codes with }k\text{ slots of }c\text{ symbols}=c^k$

Indian number names

$\text{lakh}=10^5,\text{crore}=10^7,\text{arab}=10^9,\text{kharab}=10^{11}$

International names

$\text{million}=10^6,\text{billion}=10^9,\text{trillion}=10^{12}$

How the school exam asks Power Play

The question patterns to expect, and the method that works for each.

Tricks to remember Power Play

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Multiply-Add, Divide-Subtract, Power-Times

  MA-DS-PT: Multiply powers, Add exponents; Divide powers, Subtract; Power of a power, Times.

  Use it when: Choosing the right law in a simplification.

• Negative exponent means flip

  A negative exponent sends the power to the other side of the fraction bar: $2^{-3}=\frac{1}{2^3}$.

  Use it when: Evaluating negative powers.

• Anything to the zero is one

  Divide anything by itself and you get 1, so $n^0=1$ (but never with $n=0$).

  Use it when: Zero exponent questions.

• Odd power keeps the minus

  Odd keeps the minus, even makes it vanish: $(-2{)}^5=-32$, $(-2{)}^4=16$.

  Use it when: Powers of negative numbers.

• Lakh-crore-arab: jump by 100

  Indian names jump by $10^2$ (lakh $10^5$, crore $10^7$, arab $10^9$); international names jump by $10^3$ (million $10^6$, billion $10^9$).

  Use it when: Converting between number systems.

Power Play: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Power Play

The slips examiners see most often, and what to write instead.

• ✗Mistake: Multiplying exponents when bases are multiplied

  The product and power-of-a-power rules get mixed up.

  Wrong working

  $3^4\times 3^3=3^{12}$

  Right working

  $3^4\times 3^3=3^{4+3}=3^7$; you multiply exponents only for $(3^4{)}^3=3^{12}$.

  ✓Do this: Ask: am I multiplying two powers (add) or raising a power to a power (multiply)?

  Usually costs: The whole simplification.

• ✗Mistake: Adding exponents of different bases

  The product rule is used without checking the bases match.

  Wrong working

  $2^3\times 5^2=10^5$

  Right working

  $2^3\times 5^2=8\times 25=200$. Only equal exponents combine bases: $2^3\times 5^3=10^3$.

  ✓Do this: Combine exponents only when the base is the same; combine bases only when the exponent is the same.

  Usually costs: Answer mark.

• ✗Mistake: Treating a negative exponent as a negative number

  The minus sign is read as making the value negative.

  Wrong working

  $2^{-3}=-8$

  Right working

  $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$, a positive number.

  ✓Do this: A negative exponent means a reciprocal, not a negative value.

  Usually costs: 1 mark each time.

• ✗Mistake: Coefficient outside 1 to 10 in standard form

  Students stop moving the point too early.

  Wrong working

  $5,98,53,000=59.853\times 10^6$

  Right working

  $5.9853\times 10^7$, with exactly one non-zero digit before the point.

  ✓Do this: Check $1\le x<10$ before writing the final answer.

  Usually costs: 1 mark.

• ✗Mistake: Confusing $(-3{)}^2$ with $-3^2$

  Brackets are ignored.

  Wrong working

  $-3^2=9$

  Right working

  $-3^2=-(3^2)=-9$, while $(-3{)}^2=9$.

  ✓Do this: The exponent applies only to what is directly under it; a bracket makes the minus part of the base.

  Usually costs: 1 mark.

NCERT exercise guide: Power Play

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Power Play

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   $(-2{)}^5$ equals
   (a) 10
   (b) $-10$
   (c) 32
   (d) $-32$

   Hint

   Odd power of a negative number.

   Answer

   (d) $-32$

2. Q2 · 1 mark · MCQExam level

   $10^{-3}$ equals
   (a) $-1000$
   (b) $-30$
   (c) 0.001
   (d) 0.003

   Hint

   Negative exponent means reciprocal.

   Answer

   (c) 0.001

3. Q3 · 1 mark · Very short answerExam level

   Write $5\times 5\times a\times a\times a\times b$ in exponential form.

   Hint

   Count each factor.

   Answer

   $5^2{a}^{3}b$

4. Q4 · 1 mark · Very short answerExam level

   Write $4.7\times 10^6$ in ordinary form, using Indian commas.

   Hint

   Move the point 6 places right.

   Answer

   47,00,000

5. Q5 · 2 marks · Very short answerHarder exam level

   Simplify $\frac{3^{-3}\times 3^8}{3^2}$.

   Hint

   Add, then subtract exponents.

   Answer

   $3^{-3+8-2}=3^3=27$

6. Q6 · 2 marks · Very short answerHarder exam level

   Write $4^6$ as a power of a power in two different ways, and as a power of 2. Find its value.

   Hint

   $4=2^2$.

   Answer

   $(4^2{)}^3=(4^3{)}^2=2^{12}=4096$

7. Q7 · 2 marks · Very short answerHarder exam level

   A computer password must have exactly 6 capital letters. How many passwords are possible? Give the answer in scientific notation, rounded to 3 significant figures.

   Hint

   26 choices per slot.

   Answer

   $26^6=30,89,15,776\approx 3.09\times 10^8$

8. Q8 · 3 marks · Short answerHarder exam level

   Say whether each statement is always true, sometimes true or never true, with a reason: (i) A square number is also a cube number. (ii) The product of two square numbers is a square number. (iii) $2^n$ is odd for a counting number $n$.

   Hint

   Test examples, then think in exponents.

   Answer

   (i) Sometimes: 64 is both, 4 is not a cube. (ii) Always: $a^2\times b^2=(ab{)}^2$. (iii) Never: $2^n$ has 2 as a factor, so it is even.

9. Q9 · 3 marks · Short answerHarder exam level

   Suppose a library wants a different numeric code (digits 0 to 9) for each of its 3.2 crore items. What is the smallest code length that works? How many codes would be left unused?

   Hint

   Compare 3.2 crore with powers of 10.

   Answer

   $10^7=1$ crore is too few; $10^8=10$ crore is enough, so 8 digits. Unused: $10-3.2=6.8$ crore codes ($6.8\times 10^7$).

10. Q10 · 5 marks · Long answerHarder exam level

    Suppose a culture starts with 500 bacteria and the number doubles every 20 minutes. (a) Write the number after $n$ doublings. (b) How many bacteria are there after 4 hours? Give the answer in scientific notation. (c) When were there half as many as at 4 hours?

    Hint

    4 hours has 12 periods of 20 minutes.

    Answer

    (a) $500\times 2^n$. (b) $n=12$: $500\times 4096=20,48,000=2.048\times 10^6$. (c) One doubling earlier, at 3 hours 40 minutes.

11. Q11 · 3 marks · Short answerHOTS / competency

    Show that $2^{100}$ is greater than $10^{30}$.

    Hint

    Compare $2^{10}$ with $10^3$.

    Answer

    $2^{10}=1024>1000=10^3$. Raise both to the power 10: $2^{100}=(2^{10}{)}^{10}>(10^3{)}^{10}=10^{30}$.

12. Q12 · 2 marks · Very short answerHOTS / competency

    Find $n$ if $4^n\times 8=2^{11}$.

    Hint

    Write everything as powers of 2.

    Answer

    $2^{2n}\times 2^3=2^{11}$, so $2n+3=11$ and $n=4$.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.In $7^5$, which number is the base and which the exponent?Show answer

   7 is the base and 5 is the exponent.

2. 2.State the law for $n^a\times n^b$.Show answer

   $n^a\times n^b=n^{a+b}$.

3. 3.What is $(n^a{)}^b$?Show answer

   $n^{ab}$, which also equals $(n^b{)}^a$.

4. 4.Why is $n^0=1$?Show answer

   Because $n^a÷n^a=n^0$, and any non-zero number divided by itself is 1.

5. 5.What does $10^{-3}$ equal?Show answer

   $\frac{1}{1000}=0.001$.

6. 6.What are the two conditions for scientific notation $x\times 10^y$?Show answer

   $1\le x<10$ and $y$ is an integer.

7. 7.How many 4-digit codes can be made with digits 0 to 9?Show answer

   $10^4=10,000$.

8. 8.What is the difference between linear and exponential growth?Show answer

   Linear growth adds a fixed amount each step; exponential growth multiplies by a fixed factor each step.

9. 9.Write a crore and an arab as powers of 10.Show answer

   Crore $=10^7$, arab $=10^9$.

10. 10.Is $(-1{)}^{101}$ positive or negative?Show answer

    Negative, $-1$, because 101 is odd.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you find the thickness of paper after n folds?
• ☐Can you tell the sign of a power of a negative number at a glance?
• ☐Can you write a number as a product of prime powers?
• ☐Can you use all four laws of exponents correctly?
• ☐Can you explain why $n^0=1$ and $n^{-a}=\frac{1}{n^a}$?
• ☐Can you count the codes for a lock with letters and digits?
• ☐Can you write a large and a small number in scientific notation?
• ☐Can you compare two numbers in standard form?
• ☐Can you compare $2^{30}$ and $3^{20}$ without a calculator?
• ☐Can you convert between lakh, crore, million and billion using powers of 10?

Power Play: frequently asked questions