Find out the units digit in the value of ? [Hint: ]

Understand the Question To find the units digit of , we first simplify the expression to a single power of using exponent laws: and . The units digits of powers of repeat in a predictable cycle of four values: . By dividing the final exponent by , the remainder determines the exact units digit. Step 1 · Simplify the Expression Given expression Rewrite base as Using the power of a power rule Substitute back and apply the quotient rule Step 2 · Find the Units Digit Look at the pattern of units digits for powers of The units digits repeat in a cycle of : . Divide the exponent by the cycle length A remainder of corresponds to the position in the cycle. Therefore, the units digit is . Answer > 6 Common Mistakes Dividing Exponents Directly: Incorrectly dividing the exponents directly as instead of converting bases to match first. Remainder 0 Confusion: Assuming a remainder of corresponds to . In a cyclicity of , a remainder of represents a complete cycle and corresponds to the power ().

Power Play (Exponents) | FIO

Question 4

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

Solution

Understand the Question

To find the units digit of $2^{224} \div 4^{32}$ , we first simplify the expression to a single power of $2$ using exponent laws: $(a^m)^n = a^{m \times n}$ and $a^m \div a^n = a^{m - n}$ .
The units digits of powers of $2$ repeat in a predictable cycle of four values: $2, 4, 8, 6$ .
By dividing the final exponent by $4$ , the remainder determines the exact units digit.

Step 1 · Simplify the Expression

Given expression $2^{224} \div 4^{32}$

Rewrite base $4$ as $2^2$ $4^{32} = (2^2)^{32}$

Using the power of a power rule $(a^m)^n = a^{m \times n}$

\begin{aligned} (2^2)^{32} &= 2^{2 \times 32} \\[0.6em] &= 2^{64} \end{aligned}

Substitute back and apply the quotient rule $a^m \div a^n = a^{m - n}$

\begin{aligned} 2^{224} \div 2^{64} &= 2^{224 - 64} \\[0.6em] &= 2^{160} \end{aligned}

Step 2 · Find the Units Digit

Look at the pattern of units digits for powers of $2$

$2^1 = 2$
$2^2 = 4$
$2^3 = 8$
$2^4 = 16 \rightarrow 6$
$2^5 = 32 \rightarrow 2$

The units digits repeat in a cycle of $4$ : $2, 4, 8, 6$ .

Divide the exponent $160$ by the cycle length $4$ $160 \div 4 = 40 \quad \text{with remainder } 0$

A remainder of $0$ corresponds to the $4^{\text{th}}$ position in the cycle.

Therefore, the units digit is $6$ .

Answer

6

Common Mistakes

Dividing Exponents Directly: Incorrectly dividing the exponents directly as $224 \div 32$ instead of converting bases to match first.
Remainder 0 Confusion: Assuming a remainder of $0$ corresponds to $2^0 = 1$ . In a cyclicity of $4$ , a remainder of $0$ represents a complete cycle and corresponds to the $4^{\text{th}}$ power ( $2^4 \rightarrow 6$ ).

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

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—

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

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

The worldwide population of sheep (2024) is about $10^9$ , and that of goats is also about the same. What is the total population of sheep and goats?

(i) $20^9$

(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

What was the date 1 arab/1 billion seconds ago?