Class 8 Maths Chapter -2 Figure it Out-Page No-44

🧮 NCERT Class 8 Maths Chapter 2 – Figure it Out (Page 44)

Step-by-Step Solution

Topic: Exponents and Powers
Chapter: Power Play
Book: NCERT Class 8 Mathematics (Ganita Prakash)
Question:
Find out the units digit in the value of

$$2^{224} ÷ 4^{32}$$

Hint: $4 = 2^2$

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✅ Step 1: Express 4 as a Power of 2

We know that

$$4 = 2^2$$

Therefore,

$$4^{32} = (2^2)^{32} = 2^{64}$$

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✅ Step 2: Simplify the Given Expression

Substitute $4^{32} = 2^{64}$

✅ Step 2: Simplify the Given Expression

Substitute $4^{32} = 2^{64}$

$$2^{224} ÷ 4^{32} = 2^{224} ÷ 2^{64}$$

Now, apply the law of exponents —

$$a^m ÷ a^n = a^{m-n}$$

So,

$$2^{224} ÷ 2^{64} = 2^{224 - 64} = 2^{160}$$

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✅ Step 3: Find the Units Digit of $2^{160}$

To find the units digit, let’s look at the pattern of powers of 2:

Power	Expression	Units Digit
$2^1$	2	2
$2^2$	4	4
$2^3$	8	8
$2^4$	16	6

The pattern of units digits (2, 4, 8, 6) repeats every 4 powers.

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✅ Step 4: Determine the Position in the Pattern

Now, divide the exponent 160 by 4 to find the remainder:

$$160 ÷ 4 = 40 \text{ remainder } 0$$

When the remainder is 0, the units digit corresponds to the 4th term of the pattern, which is 6.

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🎯 Final Answer

$$$$

✅ Final Answer: 6

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🧠 Key Learning Points

1. ${\mathrm{4=\; 2}}^2$ helps to simplify powers into the same base.

2. When dividing powers with the same base, subtract exponents.

3. The units digit of powers of 2 repeats every 4 steps — (2, 4, 8, 6).

4. If the exponent is a multiple of 4, the units digit will always be 6.

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