Power PlayClass 8 Mathematics NCERT Solutions
15 Solutions
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Solution 1 of 15
Q1Figure it Out (from Section 2.2)
Express the following in exponential form:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution
To Find: The exponential form of the given expressions.
Solution:
(i)
(ii)
(iii)
(iv)
(v)
or
(vi)
Final Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Q2Figure it Out (from Section 2.2)
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
Solution
To Find: The prime factorization of the given numbers in exponential form.
Solution:
(i)
648
(ii)
405
(iii)
540
(iv)
3600
Final Answer:
(i)
(ii)
(iii)
(iv)
Q3Figure it Out (from Section 2.2)
Write the numerical value of each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution
To Find: The numerical value of the given expressions.
Solution:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Final Answer:
(i)
2000
(ii)
392
(iii)
768
(iv)
225
(v)
90000
(vi)
-32,000,000
Q1Questions from Section 2.2
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 .
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Solution
Given:
Initial thickness of the paper =
The paper is folded 10 times.
Each fold doubles the thickness of the paper.
To Find:
The expression for the thickness after 10 folds.
Solution:
- Initial thickness (0 folds) =
- Thickness after 1 fold =
- Thickness after 2 folds =
- Thickness after 3 folds =
Following this pattern, the thickness after folds is .
For 10 folds, .
Therefore, the thickness after 10 folds will be .
This corresponds to option (v).
Final Answer: (v)
Q2Questions from Section 2.2
Use this observation to compute the following.
(i)
(ii)
(iii)
Solution
To Find: The value of the given exponential expressions.
Solution:
(i)
We know and .
(ii)
We know and .
(iii)
We can write as .
Final Answer:
(i)
(ii)
(iii)
Q3Questions from Section 2.2
Write the following expressions as a power of a power in at least two different ways:
(i)
(ii)
(iii)
(iv)
Solution
To Find: Express the given expressions as a power of a power using the rule .
Solution:
(i)
We need to find factors of the exponent 6. The factors are 2 and 3.
(ii)
We need to find factors of the exponent 15. The factors are 3 and 5.
(iii)
We need to find factors of the exponent 14. The factors are 2 and 7.
(iv)
We need to find factors of the exponent 8. The factors are 2 and 4.
Final Answer:
(i)
and
(ii)
and
(iii)
and
(iv)
and
Q4Questions from Section 2.2
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. On which day was the pond half full?
Solution
Given:
- The number of lotuses doubles every day.
- The pond is completely covered with lotuses on the 30th day.
To Find:
The day on which the pond was half full.
Solution:
The number of lotuses doubles each day. This means that if we go backward one day in time, the number of lotuses would be half of the current day's amount.
On the 30th day, the pond is fully covered.
To find when the pond was half full, we need to go back one day from the day it was full.
Day 30: Pond is full.
Day 29: The number of lotuses was half the number on Day 30. Therefore, the pond was half full.
Final Answer: The pond was half full on the 29th day.
Q5Questions from Section 2.2
Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. How many different ways can Roxie dress up?
Solution
Given:
Number of dresses = 7
Number of hats = 2
Number of pairs of shoes = 3
To Find:
The total number of different ways Roxie can dress up.
Solution:
To find the total number of combinations, we multiply the number of choices for each item.
For each of the 7 dresses, she can choose any of the 2 hats. This gives combinations of dresses and hats.
For each of these 14 combinations, she can choose any of the 3 pairs of shoes.
Total number of ways = (Number of dresses) (Number of hats) (Number of pairs of shoes)
Total number of ways =
Final Answer: Roxie can dress up in 42 different ways.
Q6Questions from Section 2.2
Estu says, "Next time, I will buy a lock that has 6 slots with the letters A to Z. I feel it is safer." How many passwords are possible with such a lock?
Solution
Given:
Number of slots in the lock = 6
Possible characters for each slot = Letters from A to Z.
Number of letters from A to Z = 26.
To Find:
The total number of possible passwords.
Solution:
Each of the 6 slots can be filled with any of the 26 letters. The choice for each slot is independent of the others.
- Number of choices for the 1st slot = 26
- Number of choices for the 2nd slot = 26
- Number of choices for the 3rd slot = 26
- Number of choices for the 4th slot = 26
- Number of choices for the 5th slot = 26
- Number of choices for the 6th slot = 26
Total number of passwords =
Calculating the value:
Final Answer: There are (or 308,915,776) possible passwords.
Q1Questions from Section 2.3
What is in powers of 2?
Solution
To Find: The value of in powers of 2.
Formula:
Solution:
Using the formula for division of powers with the same base:
Final Answer:
Q2Questions from Section 2.3
Can we write ?
Solution
To Verify: If the equation is true.
Formula:
Solution:
Let's simplify the right-hand side (RHS) of the equation.
RHS =
Using the rule , we have .
Substitute this into the RHS:
RHS =
Dividing by a fraction is the same as multiplying by its reciprocal:
RHS =
Now, compare with the left-hand side (LHS).
LHS =
Since LHS = RHS, the statement is true.
Final Answer: Yes, we can write .
Q3Questions from Section 2.3
Write equivalent forms of the following.
(i)
(ii)
(iii)
(iv)
(v)
Solution
To Find: Equivalent forms of the given expressions using the rule .
Solution:
(i)
(ii)
(iii)
(iv)
(v)
Final Answer:
(i)
(ii)
(iii)
(iv)
(v)
Q4Questions from Section 2.3
Simplify and write the answers in exponential form.
(i)
(ii)
(iii)
(iv)
(v)
Solution
To Find: The simplified exponential form of the given expressions.
Formulas:
Solution:
(i)
(ii)
(iii)
(iv)
Alternatively, can be written as or any non-zero number to the power of 0. Since the question asks for exponential form, we can write .
(v)
Final Answer:
(i)
(ii)
(iii)
(iv)
(v)
Q5Questions from Section 2.3
Use the power line for 7 to answer the following questions.
Solution
Given:
The power line for 7 provides these values:
To Find: The values of the given expressions.
Solution:
We will convert each number to a power of 7 and use exponent rules.
Final Answer:
Q1Questions from Section 2.4
Write these numbers in the same way: (i) 172, (ii) 5642, (iii) 6374.
Solution
To Find: Write the given numbers in expanded form using powers of 10.
Example from text:
Solution:
(i)
172
(ii)
5642
(iii)
6374
Final Answer:
(i)
(ii)
(iii)