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Question
If `(5^m xx 5^3 xx 5^-2)/5^-5`, find m.
Solution
Given, `(5^m xx 5^3 xx 5^-2)/5^-5 = 5^12`
Using laws of exponents, am + an = (a)m – n and `a^-m = 1/a^n` ...[∵ a is non-zero integer]
Then, 5m × 53 × 5–2 × 55 = 512
⇒ 5m × 58 × 5–2 = 512
⇒ 5m × 56 = 512
⇒ 5m + 6 = 1512 ...[∵ am × an = am + n]
On comparing both sides, we get
m + 6 = 12
⇒ m = 6
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am × bm = (ab)m
Simplify:
`(1/4)^-2 + (1/2)^-2 + (1/3)^-2`
Supply the missing information for diagram.
The left column of the chart lists the lengths of input chains of gold. Repeater machines are listed across the top. The other entries are the outputs you get when you send the input chain from that row through the repeater machine from that column. Copy and complete the chart.
| Input Length | Repeater Machine | ||
| × 23 | |||
| 40 | 125 | ||
| 2 | |||
| 162 | |||
Long back in ancient times, a farmer saved the life of a king’s daughter. The king decided to reward the farmer with whatever he wished. The farmer, who was a chess champion, made an unusal request:
“I would like you to place 1 rupee on the first square of my chessboard, 2 rupees on the second square, 4 on the third square, 8 on the fourth square, and so on, until you have covered all 64 squares. Each square should have twice as many rupees as the previous square.” The king thought this to be too less and asked the farmer to think of some better reward, but the farmer didn’t agree.
How much money has the farmer earned?
[Hint: The following table may help you. What is the first square on which the king will place at least Rs 10 lakh?]
| Position of Square on chess board |
Amount (in Rs) |
| 1st square | 1 |
| 2nd square | 2 |
| 3rd square | 4 |
Simplify:
`((9)^3 xx 27 xx t^4)/((3)^-2 xx (3)^4 xx t^2)`
