Write the numerical value of each of the following: (i) (ii) (iii) (iv) (v) (vi)

Power Play (Exponents) | FIO

Question 3

Write the numerical value of each of the following:

(i) $2 \times 10^3$

(ii) $7^2 \times 2^3$

(iii) $3 \times 4^4$

(iv) $(-3)^2 \times (-5)^2$

(v) $3^2 \times 10^4$

(vi) $(-2)^5 \times (-10)^6$

Solution

We will find the numerical value of each expression by expanding the powers and then performing the multiplication.

Step 1 — Calculate $2 \times 10^3$

An exponent tells us how many times to multiply a base number by itself. So, $10^3$ means we multiply 10 by itself 3 times.

$10^3 = 10 \times 10 \times 10$

$= 1000$

Now, we multiply this value by 2.

$2 \times 10^3 = 2 \times 1000$

$\boxed{2000}$

Diagram 1

Step 2 — Calculate $7^2 \times 2^3$ }

First, we find the value of $7^2$ . This means we multiply 7 by itself 2 times.

$7^2 = 7 \times 7$

$= 49$

Next, we find the value of $2^3$ . This means we multiply 2 by itself 3 times.

$2^3 = 2 \times 2 \times 2$

$= 8$

Finally, we multiply these two results together.

$7^2 \times 2^3 = 49 \times 8$

$\boxed{392}$

Step 3 — Calculate $3 \times 4^4$ }

First, we find the value of $4^4$ . This means we multiply 4 by itself 4 times.

$4^4 = 4 \times 4 \times 4 \times 4$

$= 16 \times 16$

$= 256$

Now, we multiply this value by 3.

$3 \times 4^4 = 3 \times 256$

$\boxed{768}$

Step 4 — Calculate $(-3)^2 \times (-5)^2$ }

First, we find the value of $(-3)^2$ . This means we multiply -3 by itself 2 times. Remember that a negative number multiplied by a negative number gives a positive number.

$(-3)^2 = (-3) \times (-3)$

$= 9$

Next, we find the value of $(-5)^2$ . This means we multiply -5 by itself 2 times.

$(-5)^2 = (-5) \times (-5)$

$= 25$

Finally, we multiply these two results together.

$(-3)^2 \times (-5)^2 = 9 \times 25$

$\boxed{225}$

Step 5 — Calculate $3^2 \times 10^4$ }

First, we find the value of $3^2$ . This means we multiply 3 by itself 2 times.

$3^2 = 3 \times 3$

$= 9$

Next, we find the value of $10^4$ . This means we multiply 10 by itself 4 times.

$10^4 = 10 \times 10 \times 10 \times 10$

$= 10000$

Finally, we multiply these two results together.

$3^2 \times 10^4 = 9 \times 10000$

$\boxed{90000}$

Step 6 — Calculate $(-2)^5 \times (-10)^6$ }

First, we find the value of $(-2)^5$ . This means we multiply -2 by itself 5 times. An odd power of a negative number results in a negative number.

$(-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2)$

$= 4 \times (-2) \times (-2) \times (-2)$

$= -8 \times (-2) \times (-2)$

$= 16 \times (-2)$

$= -32$

Next, we find the value of $(-10)^6$ . This means we multiply -10 by itself 6 times. An even power of a negative number results in a positive number.

$(-10)^6 = (-10) \times (-10) \times (-10) \times (-10) \times (-10) \times (-10)$

$= 100 \times 100 \times 100$

$= 10000 \times 100$

$= 1000000$

Finally, we multiply these two results together.

$(-2)^5 \times (-10)^6 = -32 \times 1000000$

$\boxed{-3200000}$

Answer

(i) 2000 (ii) 392 (iii) 768 (iv) 225 (v) 90000 (vi) -3200000

More questions in FIO

Express the following in exponential form:

(i) $6 \times 6 \times 6 \times 6$

(ii) $y \times y$

(iii) $b \times b \times b \times b$

(iv) $5 \times 5 \times 7 \times 7 \times 7$

(v) $2 \times 2 \times a \times a$

(vi) $a \times a \times a \times c \times c \times c \times c \times d$

Express each of the following as a product of powers of their prime factors in exponential form:

(i) 648

(ii) 405

(iii) 540

(iv) 3600

Write the numerical value of each of the following:

(i) $2 \times 10^3$

(ii) $7^2 \times 2^3$

(iii) $3 \times 4^4$

(iv) $(-3)^2 \times (-5)^2$

(v) $3^2 \times 10^4$

(vi) $(-2)^5 \times (-10)^6$

Find out the units digit in the value of $2^{224} \div 4^{32}$ ? [Hint: $4 = 2^2$ ]

There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would there be after 40 days?

Write the given number as the product of two or more powers in three different ways. The powers can be any integers.

(i) $64^3$ (ii) $192^8$ (iii) $32^{-5}$

Examine each statement below and find out if it is 'Always True', 'Only Sometimes True', or 'Never True'. Explain your reasoning.

(i) Cube numbers are also square numbers.

(ii) Fourth powers are also square numbers.

(iii) The fifth power of a number is divisible by the cube of that number.

(iv) The product of two cube numbers is a cube number.

(v) $q^{46}$ is both a 4th power and a 6th power ( $q$ is a prime number).

Simplify and write these in the exponential form.

(i) $10^{-2} \times 10^{-5}$

(ii) $5^7 \div 5^4$

(iii) $9^{-7} \div 9^4$

(iv) $(13^{-2})^{-3}$

(v) $m^5n^{12}(mn)^9$

If $12^2 = 144$ what is

(i) $(1.2)^2$

(ii) $(0.12)^2$

(iii) $(0.012)^2$

(iv) $120^2$

Q10

Circle the numbers that are the same—

$2^4 \times 3^6$ $6^4 \times 3^2$ $6^{10}$ $18^2 \times 6^2$ $6^{24}$

Q11

Identify the greater number in each of the following—

(i) $4^3$ or $3^4$

(ii) $2^8$ or $8^2$

(iii) $100^2$ or $2^{100}$

Q12

A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits 0–9, how many digits should the code consist of?

Q13

64 is a square number ( $8^2$ ) and a cube number ( $4^3$ ). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Q14

A digital locker has an alphanumeric (it can have both digits and letters) passcode of length 5. Some example codes are G89P0, 38098, BRJKW, and 003AZ. How many such codes are possible?

Q15

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(ii) $20^9$

(ii) $10^{11}$

(iii) $10^{10}$

(iv) $10^{18}$

(v) $2 \times 10^9$

(vi) $10^9 + 10^9$

Q16

Calculate and write the answer in scientific notation:

(i) If each person in the world had 30 pieces of clothing, find the total number of pieces of clothing.

(ii) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees.

(iii) The human body has about 38 trillion bacterial cells. Find the bacterial population residing in all humans in the world.

(iv) Total time spent eating in a lifetime in seconds.

Q17

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