Question 4 - Figure it out - Page 44, 45 - Chapter 2 Class 8 - Power Play (Ganita Prakash)

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Question 4 (i) 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.Cube numbers are numbers like 23 Square numbers are numbers like 42 For numbers 13 and 12 13 = 1 12 = 1 Here, both cube and square numbers are equal For numbers 23 and 42 23 = 8 42 = 16 Here, both cube and square numbers are not equal For numbers 43 and 82 43 = 64 82 = 64 Here, both cube and square numbers are equal Note: This is only true for numbers that are perfect sixth powers, like n6 = (n3)2 = (n2)3 Since the given statement is sometimes true, sometimes false. Therefore, given statement is Only Sometimes True Question 4 (ii) Examine each statement below and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning. (ii) Fourth powers are also square numbers.Fourth powers are like 𝑛^4 And, we can write 𝒏^𝟒=(𝒏^𝟐 )^𝟐 Thus, we can write all fourth powers as square numbers Example: 16 = 24 16 = 42 Thus, given statement is Always True Question 4 (iii) Examine each statement below and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning. (iii) The fifth power of a number is divisible by the cube of that number.Basically the statement is 𝒏^𝟓 is divisible by 𝒏^𝟑 This is always true, as 𝒏^𝟓/𝒏^𝟑 =𝒏^(𝟓−𝟑) = 𝒏^𝟐 Since n is an integer, n2 is also an integer, meaning the division is exact. Example: 25 = 32 23 = 8 Now, 𝟐^𝟓/𝟐^𝟑 =𝟑𝟐/𝟖 =𝟒 Since 4 is an integer, it means fully divisible Thus, given statement is Always True Question 4 (iv) Examine each statement below and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning. (iv) The product of two cube numbers is a cube number.Let the two cube numbers be 𝒂^𝟑 and 𝒃^𝟑 Now, product of this cube number is 𝒂^𝟑 × 𝒃^𝟑=(𝒂 × 𝒃)^𝟑 =(𝒂𝒃)^𝟑 And, (𝒂𝒃)^𝟑 is a cube number Thus, given statement is Always True Example: Let’s take 23 and 33 Now, their product 𝟐^𝟑 × 𝟑^𝟑=(𝟐 × 𝟑)^𝟑 =𝟔^𝟑 Since 𝟔^𝟑 is a cube number, our statement is Always True Question 4 (v) Examine each statement below and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning. (v) 𝑞^46 is both a 4^𝑡ℎ power and a 6th power (q is a prime number).Now, For 𝑞^46 to be a 4th power, the exponent 46 must be divisible by 4. It is not. For 𝑞^46 to be a 4th power, 46 must be divisible by 6. It is not. Thus, given statement is Never True