Zero cannot be the base for zero or negative exponents. Step 1 — The rule for any non-zero number raised to the power of 0 Let us recall the division rule for exponents. When we divide powers with the same base, we subtract the exponents. Let's consider a special case where the exponents are the same. So, let . We also know that any non-zero number divided by itself is 1. So, we conclude that . This rule is only true if is not 0. This is because we cannot divide by zero. So, for any . Step 2 — The rule for zero raised to a positive power Now, let us consider what happens when the base is 0. If we have , where is a positive whole number. For example, means . Similarly, means . Multiplying 0 by itself any positive number of times always gives 0. So, for any . 0.> Step 3 — Why is undefined Let's try to find the value of . From Step 1, suggests should be 1. From Step 2, suggests should be 0. But 1 is not equal to 0. So, cannot be both 1 and 0. Because of this conflict, is undefined in basic algebra. So, cannot be 0 if it is the base with a zero exponent. Step 4 — Why is undefined Let us recall the rule for negative exponents. The rule for negative exponents is . This rule also requires that is not 0. Let's try to use this rule with a base of 0. Suppose we want to find . Using the rule, . From Step 2, we know that . So, . Division by zero is not allowed in mathematics. Therefore, any expression like is also undefined. Thus, cannot be 0 if it is the base for a negative exponent. Answer > (i) cannot be the base when the exponent is 0. This is because is undefined. > (ii) cannot be the base when the exponent is negative. This leads to division by zero. > (iii) If is the exponent, it can be 0. This is true if the base is not 0 (e.g., ).

Power Play (Exponents) | IT

Question 19

Why can't $n$ be 0?

Solution

Zero cannot be the base for zero or negative exponents.

Step 1 — The rule for any non-zero number raised to the power of 0

Let us recall the division rule for exponents. When we divide powers with the same base, we subtract the exponents. $a^m \div a^n = a^{m-n}$ Let's consider a special case where the exponents are the same. So, let $m=n$ . $a^m \div a^m = a^{m-m}$ $a^m \div a^m = a^0$ We also know that any non-zero number divided by itself is 1. $a^m \div a^m = 1$ So, we conclude that $a^0 = 1$ . This rule is only true if $a$ is not 0. This is because we cannot divide by zero. So, $a^0 = 1$ for any $a \neq 0$ .

Diagram 1

Step 2 — The rule for zero raised to a positive power

Now, let us consider what happens when the base is 0. If we have $0^n$ , where $n$ is a positive whole number. For example, $0^2$ means $0 \times 0$ . $0^2 = 0$ Similarly, $0^3$ means $0 \times 0 \times 0$ . $0^3 = 0$ Multiplying 0 by itself any positive number of times always gives 0. So, $0^n = 0$ for any $n > 0$ .

<DIAGRAM: A sequence of calculations: 0^1=0, 0^2=0, 0^3=0. An arrow points to a general rule: 0^n=0 for n>0.>

Step 3 — Why $0^0$ is undefined

Let's try to find the value of $0^0$ . From Step 1, $a^0 = 1$ suggests $0^0$ should be 1. From Step 2, $0^n = 0$ suggests $0^0$ should be 0. But 1 is not equal to 0. So, $0^0$ cannot be both 1 and 0. Because of this conflict, $0^0$ is undefined in basic algebra. So, $n$ cannot be 0 if it is the base with a zero exponent.

Diagram 2

Step 4 — Why $0^{\text{negative exponent}}$ is undefined

Let us recall the rule for negative exponents. The rule for negative exponents is $a^{-n} = \frac{1}{a^n}$ . This rule also requires that $a$ is not 0. Let's try to use this rule with a base of 0. Suppose we want to find $0^{-2}$ . Using the rule, $0^{-2} = \frac{1}{0^2}$ . From Step 2, we know that $0^2 = 0$ . So, $0^{-2} = \frac{1}{0}$ . Division by zero is not allowed in mathematics. Therefore, any expression like $0^{\text{negative exponent}}$ is also undefined. Thus, $n$ cannot be 0 if it is the base for a negative exponent.

Diagram 3

Answer

(i) $n$ cannot be the base when the exponent is 0. This is because $0^0$ is undefined. (ii) $n$ cannot be the base when the exponent is negative. This leads to division by zero. (iii) If $n$ is the exponent, it can be 0. This is true if the base is not 0 (e.g., $5^0=1$ ).

More questions in IT

Say you can fold a sheet of paper as many times as you wish. What would its thickness be after 30 folds? Make a guess.

Now, what do you think the thickness would be after 30 folds? 45 folds? Make a guess.

Fill the table below.

Notice the change in thickness after two folds. By how much does it increase?

After any 3 folds, the thickness increases 8 times (= 2 × 2 × 2). Check if that is true.

Which expression describes the thickness of a sheet of paper after it is folded 10 times? The initial thickness is represented by the letter-number $v$ .

(i) $10v$

(ii) $10 + v$

(iii) $2 \times 10 \times v$

(iv) $2^{10}$

(v) $2^{10}v$

(vi) $10^2v$

What is $(-1)^5$ ? Is it positive or negative? What about $(-1)^{56}$ ?

Is $(-2)^4 = 16$ ? Verify.

What is $0^2$ , $0^5$ ? What is $0^n$ ?

Q10

$3^7$ can also be written as $3^2 \times 3^5$ . Can you reason out why?

Q11

Use this observation to compute the following:

(i) $2^9$ (ii) $5^7$ (iii) $4^6$

Q12

Write the following expressions as a power of a power in at least two different ways:

(i) $8^6$

(ii) $7^{15}$

(iii) $9^{14}$

(iv) $5^8$

Q13

Context: In the middle of a beautiful, magical pond lies a bright pink lotus. The number of lotuses doubles every day in this pond. After 30 days, the pond is completely covered with lotuses.

Q. Write the number of lotuses (in exponential form) when the pond was —

(i) fully covered (ii) half covered

Q14

Use the observation $m^a \times n^a = (mn)^a$ to compute the value of $2^5 \times 5^5$ .

Q15

Simplify $\frac{10^4}{5^4}$ and write it in exponential form.

Q16

Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. How many different ways can Roxie dress up?

Hint: Try drawing a diagram like the one above.

Q17

Context: Estu says, "Next time, I will buy a lock that has 6 slots with the letters A to Z. I feel it is safer."

Q. How many passwords are possible with such a lock?

Q18

What is $2^{100} \div 2^{25}$ in powers of 2?

Q19

Why can't $n$ be 0?

Q20

Can we write $10^3 = \frac{1}{10^{-3}}$ ?

Q21

We had required $a$ and $b$ to be counting numbers. Can $a$ and $b$ be any integers? Will the generalised forms still hold true?

Q22

Write equivalent forms of the following.

(i) $2^{-4}$

(ii) $10^{-5}$

(iii) $(-7)^{-2}$

(iv) $(-5)^{-3}$

(v) $10^{-100}$

Q23

Simplify and write the answers in exponential form.

(i) $2^{-4} \times 2^7$

(ii) $3^2 \times 3^{-5} \times 3^6$

(iii) $p^3 \times p^{-10}$

(iv) $2^4 \times (-4)^{-2}$

(v) $8^p \times 8^q$

Q24

How many times larger than $4^{-2}$ is $4^2$ ?

Q25

Use the power line for 7 to answer the following questions.

Q26

Write these numbers in the same way:

(i) 172 (ii) 5642 (iii) 6374

Q27

Context: (i) The Sun is located 30,00,00,00,00,00,00,00,00,00,00,000 m from the centre of our Milky Way galaxy. (ii) The number of stars in our galaxy is 1,00,00,00,00,000. (iii) The mass of the Earth is 59,76,00,00,00,00,00,00,00,00,00,00,000 kg.

Q. Write the large-number facts we read just before in this form.

Q28

Context: The distance between the Sun and Saturn is $14,33,50,00,00,000\text{ m} = 1.4335 \times 10^{12}\text{ m}$ . The distance between Saturn and Uranus is $14,39,00,00,00,000\text{ m} = 1.439 \times 10^{12}\text{ m}$ . The distance between the Sun and Earth is $1,49,60,00,00,000\text{ m} = 1.496 \times 10^{11}\text{ m}$ .

Q. Can you say which of the three distances is the smallest?

Q29

The number line below shows the distance between the Sun and Saturn ( $1.4335 \times 10^{12}\text{ m}$ ). On the number line below, mark the relative position of the Earth. The distance between the Sun and the Earth is $1.496 \times 10^{11}\text{ m}$ .

Q30

Express the following numbers in standard form.

(i) 59,853

(ii) 65,950

(iii) 34,30,000

(iv) 70,04,00,00,000

Q31

Roxie wonders, “Instead of jaggery if we use 1-rupee coins, how many coins are needed to equal my weight?”. How can we find out?

Q32

Context: Roxie wonders, "Instead of jaggery if we use 1-rupee coins, how many coins are needed to equal my weight?".

Q. Would the number of coins be in hundreds, thousands, lakhs, crores, or even more? Make an instinctive guess.

Q33

Context: Roxie wonders, "Instead of jaggery if we use 1-rupee coins, how many coins are needed to equal my weight?".

Q. Find the answer by making necessary and reasonable assumptions and approximations for the unknowns. Remember, we are not looking for an exact answer but a reasonably close estimate.

How about measuring to find out the weight of a 1-rupee coin?

Q34

Context: Estu asks, "What if we use 5-rupee coins or 10-rupee notes instead? How much money could it be?".

Q. Make an instinctive guess first. Then find out (make necessary and reasonable assumptions about the unknown details and find the answers).

Q35

Context: Estu says, "When I become an adult, I would like to donate notebooks worth my weight every year". Roxie says, "When I grow up, I would like to do annadāna (offering grains or meals) worth my weight every year".

Q. How many people might benefit from each of these offerings in a year? Again, guess first before finding out.

Q36

Context: Roxie and Estu overheard someone saying—"We did padayātra for about 400 km to reach this place! We arrived early this morning."

Q. How long ago would they have started their journey?

Q37

Context: Roxie and Estu overheard someone saying—"We did padayātra for about 400 km to reach this place! We arrived early this morning."

Q. Find answers by making necessary assumptions and approximations. Do guess first before calculating to check how close your guess was!

Q38

How many times can a person circumnavigate (go around the world) the Earth in their lifetime if they walk non-stop? Consider the distance around the Earth as 40,000 km.

Q39

Context: Roxie tells Estu about a science-fiction novel she is reading where they build a ladder to reach the moon, "... I wonder if we actually had a ladder like that, how many steps would it have?".

Q. What do you think? Make an instinctive guess first.

Q40

Q. Would the number of steps be in thousands, lakhs, crores, or even more?

Q41

Can you come up with some examples of linear growth and of exponential growth?

Q42

With a global human population of about $8 \times 10^9$ and about $4 \times 10^5$ African elephants, can we say that there are nearly 20,000 people for every African elephant?

Q43

Calculate and write the answer using scientific notation:

(i) How many ants are there for every human in the world?

(ii) If a flock of starlings contains 10,000 birds, how many flocks could there be in the world?

(iii) If each tree had about $10^4$ leaves, find the total number of leaves on all the trees in the world.

(iv) If you stacked sheets of paper on top of each other, how many would you need to reach the Moon?

Q44

A different way to say your age!

"How old are you?" asked Estu. "I completed 13 years a few weeks ago!" said Roxie. "How old are you?" asked Estu again. "I'm 4840 days old today!" said Roxie. "How old are you?" asked Estu again. "I'm ______ hours old!" said Roxie.

Make an estimate before finding this number.

Estu: "I am 4070 days old today. Can you find out my date of birth?"

Q45

If you have lived for a million seconds, how old would you be?

Q46

$10^5$ seconds $\approx 1.16$ days and $10^6$ seconds $\approx 11.57$ days. Think of some events or phenomena whose time is of the order of: (i) $10^5$ seconds (ii) $10^6$ seconds

Write them in scientific notation.

Q47

A fossil of Kelenken Guillermoi, a type of terror bird, is dated to 15 million years ago.

Q48

Plants on land started 47 crore/470 million years ago ( $\approx$ _________________ seconds).

Q49

Calculate and write the answer using scientific notation:

(i) If one star is counted every second, how long would it take to count all the stars in the universe? Answer in terms of the number of seconds using scientific notation. (ii) If one could drink a glass of water (200 ml) every 10 seconds, how long would it take to finish the entire volume of water on Earth?

Q50

Context: Observe the names million ( $10^6$ ), billion ( $10^9$ ), trillion ( $10^{12}$ ), quadrillion ( $10^{15}$ ), quintillion ( $10^{18}$ ), sextillion ( $10^{21}$ ), septillion ( $10^{24}$ ), octillion ( $10^{27}$ ), nonillion ( $10^{30}$ ), decillion ( $10^{33}$ ).

Q. What does the first part of each name denote?