GSEB Class 8 Maths Solutions Chapter 7 ઘન અને ઘનમૂળ Exercise 7.2

 

Question 1. નીચે આપેલી દરેક સંખ્યાનું અવિભાજ્ય અવયવીકરણની રીતે શોધોઃ
(i) 64
Answer: 64ના અવિભાજ્ય અવયવો મેળવીએ.

\( 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)
\( \implies \sqrt[3]{64} = 2 \times 2 \)
\( = 4 \)
આમ, 64નું ઘનમૂળ 4 છે.
In simple words: First, find the prime factors of 64 by dividing it repeatedly by 2. Then, group these factors in threes. Since we have three groups of two, multiply one factor from each group to get the cube root. The cube root of 64 is 4.

Exam Tip: Remember to group prime factors into triplets when finding the cube root. If any factor is left over, the number is not a perfect cube.

 

Question 1. (ii) 512
Answer: 512ના અવિભાજ્ય અવયવો મેળવીએ.

\( 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)
\( \implies \sqrt[3]{512} = 2 \times 2 \times 2 \)
\( = 8 \)
આમ, 512નું ઘનમૂળ 8 છે.
In simple words: To find the cube root of 512, perform prime factorization. You will get nine 2s. Group these 2s into sets of three. Taking one 2 from each group and multiplying them gives the cube root, which is 8.

Exam Tip: For larger numbers, prime factorization helps break down the problem into smaller, manageable steps, making cube root calculation easier.

 

Question 1. (iii) 10648
Answer: 10648ના અવિભાજ્ય અવયવો મેળવીએ.

\( 10648 = 2 \times 2 \times 2 \times 11 \times 11 \times 11 \)
\( \implies \sqrt[3]{10648} = 2 \times 11 \)
\( = 22 \)
આમ, 10648નું ઘનમૂળ 22 છે.
In simple words: To discover the cube root of 10648, use prime factorization. You'll obtain three 2s and three 11s. Pair them up in groups of three. Multiply one factor from each group to get 22, which is the cube root.

Exam Tip: When doing prime factorization, start with the smallest prime numbers and work your way up. This makes the process more systematic and helps prevent errors.

 

Question 1. (iv) 27000
Answer: 27000ના અવિભાજ્ય અવયવો મેળવીએ.

\( 27000 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 \)
\( \implies \sqrt[3]{27000} = 2 \times 3 \times 5 \)
\( = 30 \)
આમ, 27000નું ઘનમૂળ 30 છે.
In simple words: To determine the cube root of 27000, find its prime factors. You will notice three 2s, three 3s, and three 5s. Group these factors in threes. Take one factor from each group and multiply them together to get 30, which is the cube root.

Exam Tip: When a number ends in zero, you can often factor out 10 or 2 and 5 initially, simplifying the subsequent factorization steps.

 

Question 1. (v) 15625
Answer: 15625ના અવિભાજ્ય અવયવો મેળવીએ.

\( 15625 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \)
\( \implies \sqrt[3]{15625} = 5 \times 5 \)
\( = 25 \)
આમ, 15625નું ઘનમૂળ 25 છે.
In simple words: To find the cube root of 15625, perform prime factorization. You will get six 5s. Group these 5s into two sets of three. Taking one 5 from each group and multiplying them gives the cube root, which is 25.

Exam Tip: Numbers ending in 5 are always divisible by 5. This divisibility rule helps speed up prime factorization for such numbers.

 

Question 1. (vi) 13824
Answer: 13824ના અવિભાજ્ય અવયવો મેળવીએ.

\( 13824 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \)
\( \implies \sqrt[3]{13824} = 2 \times 2 \times 2 \times 3 \)
\( = 24 \)
આમ, 13824નું ઘનમૂળ 24 છે.
In simple words: To find the cube root of 13824, start by finding its prime factors. You'll observe nine 2s and three 3s. Group these factors in sets of three. Multiply one factor from each group to get 24, which is the cube root.

Exam Tip: Be careful to count all prime factors correctly and ensure they form complete triplets for perfect cubes. Any leftover factors indicate it's not a perfect cube.

 

Question 1. (vii) 110592
Answer: 110592ના અવિભાજ્ય અવયવો મેળવીએ.

\( 110592 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \)
\( \implies \sqrt[3]{110592} = 2 \times 2 \times 2 \times 2 \times 3 \)
\( = 48 \)
આમ, 110592નું ઘનમૂળ 48 છે.
In simple words: To get the cube root of 110592, do its prime factorization. You will get twelve 2s and three 3s. Group these factors in sets of three. Take one factor from each group and multiply them to find the cube root, which is 48.

Exam Tip: For larger numbers like this, neatly organizing your prime factorization steps helps avoid counting errors and ensures an accurate result.

 

Question 1. (viii) 46656
Answer: 46656ના અવિભાજ્ય અવયવો મેળવીએ.

\( 46656 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)
\( \implies \sqrt[3]{46656} = 2 \times 2 \times 3 \times 3 \)
\( = 36 \)
આમ, 46656નું ઘનમૂળ 36 છે.
In simple words: To determine the cube root of 46656, perform prime factorization. You will get six 2s and six 3s. Group these factors in sets of three. Take one factor from each group and multiply them to find the cube root, which is 36.

Exam Tip: For numbers with many prime factors, double-check your grouping into triplets. A single miscount can lead to an incorrect cube root.

 

Question 1. (ix) 175616
Answer: 175616ના અવિભાજ્ય અવયવો મેળવીએ.

\( 175616 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 7 \)
\( \implies \sqrt[3]{175616} = 2 \times 2 \times 2 \times 7 \)
\( = 56 \)
આમ, 175616નું ઘનમૂળ 56 છે.
In simple words: To discover the cube root of 175616, perform prime factorization. You will notice nine 2s and three 7s. Group these factors in sets of three. Take one factor from each group and multiply them to find the cube root, which is 56.

Exam Tip: Recognizing prime numbers like 7 and common cubes (like \( 7^3 = 343 \)) can help in quicker factorization and cube root determination.

 

Question 1. (x) 91125
Answer: 91125ના અવિભાજ્ય અવયવો મેળવીએ.

\( 91125 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 \)
\( \implies \sqrt[3]{91125} = 3 \times 3 \times 5 \)
\( = 45 \)
આમ, 91125નું ઘનમૂળ 45 છે.
In simple words: To get the cube root of 91125, find its prime factors. You will have six 3s and three 5s. Group these factors in sets of three. Multiply one factor from each group to find the cube root, which is 45.

Exam Tip: When dealing with numbers divisible by both 3 and 5, remember the divisibility rules: a number is divisible by 3 if the sum of its digits is divisible by 3, and by 5 if it ends in 0 or 5.

 

Question 2. નીચેનું વિધાન ખરું છે કે ખોટું તે કહોઃ
(i) કોઈ પણ એકી સંખ્યાનો ધન બેકી સંખ્યા હોય.
Answer: ખોટું, કારણ કે એકી સંખ્યાનો ધન હંમેશાં એકી સંખ્યા જ હોય.
In simple words: False, because when you cube an odd number, the result is always an odd number.

Exam Tip: Remember that the cube of an odd number is always odd, and the cube of an even number is always even. This rule holds true for all integers.

 

Question 2. (ii) પૂર્ણઘન સંખ્યાના અંતિમ બે અંકો શૂન્ય ન હોય.
Answer: ખરું, કારણ કે જેના અંતિમ અંકો શૂન્ય હોય તેવી પૂર્ણઘન સંખ્યાને હંમેશાં ત્રણ કે ત્રણના ગુણકમાં શૂન્યો હોય.
In simple words: True, because for a perfect cube to end in zeros, it must have groups of three zeros at the end, not just two.

Exam Tip: A perfect cube ending in zeros must have its number of zeros as a multiple of 3 (e.g., 1000 has three zeros, \( 10^3 \)). If a number ends with two zeros (e.g., 100, 400), it cannot be a perfect cube.

 

Question 2. (iii) જો કોઈ સંખ્યાનો વર્ગ કરતાં એકમનો અંક 5 આવે, તો ધન કરતાં મળતી સંખ્યાના છેલ્લા બે અંક 25 આવે.
Answer: ખોટું, આવું હંમેશાં ન હોય. જુઓ \( 15^2 = 225 \) અને \( 15^3 = 3375 \).
In simple words: False, this is not always true. For example, the square of 15 is 225, but its cube is 3375, where the last two digits are not 25.

Exam Tip: Do not assume a pattern based on a few examples; always check for counterexamples or general mathematical principles. The rule for cubes ending in 5 requires the last *three* digits to be 125, 375, 625, or 875, not necessarily 25.

 

Question 2. (iv) એવી કોઈ પૂર્ણઘન સંખ્યા ના મળે કે જેનો એકમનો અંક 8 હોય.
Answer: ખોટું, જુઓ \( 8 = 2^3 \), \( 1728 = 12^3 \).
In simple words: False, this statement is incorrect. For instance, the number 8 is a perfect cube (\( 2^3 \)), and its unit digit is 8. Also, 1728 (\( 12^3 \)) has a unit digit of 8.

Exam Tip: Remember the unit digits of cubes: 1 cube ends in 1, 2 cube ends in 8, 3 cube ends in 7, 4 cube ends in 4, 5 cube ends in 5, 6 cube ends in 6, 7 cube ends in 3, 8 cube ends in 2, 9 cube ends in 9, 0 cube ends in 0. The unit digit 8 is possible for cubes.

 

Question 2. (v) બે અંકોવાળી સંખ્યાનો ધન કરતાં મળતી સંખ્યા ત્રણ અંકોની પણ હોય.
Answer: ખોટું, જુઓ \( 10^3 = 1000 \); ધન ચાર અંકોની સંખ્યા જ હોય.
In simple words: False, this is wrong. The cube of the smallest two-digit number, 10, is 1000, which has four digits. So, the cube of any two-digit number will always have four or more digits.

Exam Tip: The smallest two-digit number is 10 (\( 10^3 = 1000 \)) and the largest is 99 (\( 99^3 = 970299 \)). Use these boundary cases to test statements about the number of digits in cubes.

 

Question 2. (vi) બે અંકોવાળી સંખ્યાનો ધન કરતાં સાત કે તેથી વધુ અંકોની સંખ્યા પણ મળે.
Answer: ખોટું, જુઓ \( 99^3 = 970299 \); છ અંકોની સંખ્યા જ હોય.
In simple words: False, this is incorrect. The cube of the largest two-digit number, 99, is 970299, which has six digits. Therefore, the cube of a two-digit number will never have seven or more digits.

Exam Tip: To verify statements about the maximum number of digits, always test the largest number in the given range (e.g., 99 for two-digit numbers).

 

Question 2. (vii) એક અંકની સંખ્યાનો ધન કરવાથી એક અંકની સંખ્યા પણ મળે.
Answer: ખરું, જુઓ \( 1^3 = 1 \) તથા \( 2^3 = 8 \).
In simple words: True, this statement is correct. When you cube a single-digit number, the result can also be a single-digit number, as seen with \( 1^3 \) which is 1, and \( 2^3 \) which is 8.

Exam Tip: Be familiar with the cubes of single-digit numbers (1 to 9). This helps in quickly determining the unit digit of a cube or checking the range of cube values.

 

Question 3. જો તમને એમ કહેવામાં આવે કે 1331 એ પૂર્ણઘન સંખ્યા છે. શું તમે અવિભાજ્ય અવયવીકરણની પ્રક્રિયા વિના જ તેનું ઘનમૂળ શોધી શકો? તેવી જ રીતે 4913; 12187; 32768ના ઘનમૂળનું અનુમાન કરો.
Answer: 1331 સંખ્યાને બે વિભાગમાં વહેંચીએ.
1331 \( \implies \) 1 અને 331
પહેલો વિભાગ 331ના એકમનો અંક 1 છે.
\( \therefore \) 1331ના ઘનમૂળનો એકમનો અંક 1 હોય. (\( \because 1^3 = 1 \) અને \( \sqrt[3]{1} = 1 \))
હવે બીજો વિભાગ 1 છે.
\( \therefore \) 1નું ઘનમૂળ 1 છે.
આમ, 1331નું ઘનમૂળ 11 છે. (\( \therefore \sqrt[3]{1331} = 11 \))
(i) 4913 સંખ્યાને બે વિભાગમાં વહેંચીએ.
4913 \( \implies \) 4 અને 913
પહેલો વિભાગ 913ના એકમનો અંક 3 છે.
\( \therefore \) 4913ના ઘનમૂળનો એકમનો અંક 7 હોય (\( \because 7^3 = 343 \) અને \( \sqrt[3]{343} = 7 \))
હવે બીજો વિભાગ 4 છે.
\( 1^3 = 1 \) અને \( 2^3 = 8 \) અહીં, \( 1 < 4 < 8 \)
\( \therefore 1^3 < 4 < 2^3 \)
\( \therefore \) 4913ના ઘનમૂળનો દશકનો અંક 1 છે.
આમ, 4913નું ઘનમૂળ 17 છે. (\( \therefore \sqrt[3]{4913} = 17 \))
(ii) 12167
12167ને બે વિભાગમાં વહેંચીએ.
12167 \( \implies \) 12 અને 167
પહેલો વિભાગ 167ના એકમનો અંક 7 છે.
\( \therefore \) 12167ના ઘનમૂળનો એકમનો અંક 3 હોય. (\( \because 3^3 = 27 \))
હવે બીજો વિભાગ 12 છે.
\( 2^3 = 8 \) અને \( 3^3 = 27 \)
\( 8 < 12 < 27 \)
એટલે કે \( 2^3 < 12 < 3^3 \)
\( \therefore \) 12167ના ઘનમૂળનો દશકનો અંક 2 હોય.
આમ, 12167નું ઘનમૂળ 23 છે. (\( \therefore \sqrt[3]{12167} = 23 \))
(iii) 32768 સંખ્યાને બે વિભાગમાં વહેંચીએ.
32768 \( \implies \) 32 અને 768
પહેલો વિભાગ 768ના એકમનો અંક 8 છે.
\( \therefore \) 32768ના ઘનમૂળનો એકમનો અંક 2 હોય. (\( \because 2^3 = 8 \) અને \( \sqrt[3]{8} = 2 \))
હવે બીજો વિભાગ 32 છે.
\( 3^3 = 27 \) અને \( 4^3 = 64 \)
અહીં, \( 27 < 32 < 64 \)
\( \therefore 3^3 < 32 < 4^3 \)
\( \therefore \) 32768ના ઘનમૂળનો દશકનો અંક 3 હોય.
આમ, 32768નું ઘનમૂળ 32 છે. (\( \therefore \sqrt[3]{32768} = 32 \))
In simple words: To estimate the cube root without prime factorization, split the number into two parts. The last three digits determine the unit digit of the cube root. The remaining digits determine the tens digit of the cube root by finding which two perfect cubes it falls between. For 1331, 331 indicates a unit digit of 1, and 1 indicates a tens digit of 1, so the cube root is 11. Apply this process to 4913, 12187, and 32768 to find their cube roots.

Exam Tip: This estimation method is effective for perfect cubes. Memorize the cubes of single-digit numbers (1-9) as a crucial tool for quickly identifying the unit digit and estimating the tens digit.