GSEB Class 7 Maths Solutions Chapter 13 ઘાત અને ઘાતાંક Exercise 13.1

1. કિંમત શોધો:

 

Question 1. (i) \( 2^6 \)
Answer: \( 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \)
In simple words: To find the value, multiply the base number (2) by itself as many times as the power (6). This means doing \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \), which gives the result of 64.

Exam Tip: Remember that a power indicates repeated multiplication of the base number. Avoid adding the numbers instead of multiplying.

 

Question 1. (ii) \( 9^3 \)
Answer: \( 9^3 = 9 \times 9 \times 9 = 729 \)
In simple words: To find the value, you multiply the base number (9) by itself three times, as indicated by the power. So, \( 9 \times 9 = 81 \), and then \( 81 \times 9 = 729 \).

Exam Tip: Practice multiplication tables to quickly calculate powers of smaller numbers. For larger numbers, perform step-by-step multiplication.

 

Question 1. (iii) \( 11^2 \)
Answer: \( 11^2 = 11 \times 11 = 121 \)
In simple words: To find the value, multiply the base number (11) by itself two times, as shown by the power. So, \( 11 \times 11 \) equals 121.

Exam Tip: Squaring a number means multiplying it by itself. Knowing common squares (like \( 11^2 = 121 \)) can save time in exams.

 

Question 1. (iv) \( 5^4 \)
Answer: \( 5^4 = 5 \times 5 \times 5 \times 5 = 625 \)
In simple words: To find the value, multiply the base number (5) by itself four times, as the power shows. This gives \( 5 \times 5 = 25 \), then \( 25 \times 5 = 125 \), and finally \( 125 \times 5 = 625 \).

Exam Tip: When dealing with higher powers, compute step-by-step to avoid errors, for example, \( 5^4 = (5^2) \times (5^2) = 25 \times 25 \).

 

Question 2. નીચેના દરેકને ઘાત સ્વરૂપને ઘાત સ્વરૂપે દર્શાવો:
(i) \( 6 \times 6 \times 6 \times 6 \)
(ii) \( t \times t \)
(iii) \( b \times b \times b \times b \)
(iv) \( 5 \times 5 \times 7 \times 7 \times 7 \)
(v) \( 2 \times 2 \times a \times a \)
(vi) \( a \times a \times a \times c \times c \times c \times d \)
Answer:
(i) \( 6 \times 6 \times 6 \times 6 = 6^4 \)
(ii) \( t \times t = t^2 \)
(iii) \( b \times b \times b \times b = b^4 \)
(iv) \( 5 \times 5 \times 7 \times 7 \times 7 = 5^2 \times 7^3 \)
(v) \( 2 \times 2 \times a \times a = 2^2 \times a^2 \)
(vi) \( a \times a \times a \times c \times c \times c \times d = a^3 \times c^3 \times d \)
In simple words: To write an expression in exponential form, count how many times each base number or variable is multiplied by itself. That count becomes the power, or exponent, for that base. If different bases are multiplied, write each in exponential form and multiply them together.

Exam Tip: The exponent indicates the number of times the base is multiplied by itself. A variable without an explicit exponent (like d) implies an exponent of 1.

 

Question 3. નીચે દર્શાવેલ દરેક સંખ્યાને ઘાત સ્વરૂપે ઘાતાંક સંકેતનો ઉપયોગ કરીને લખો:

 

Question 3. (i) 512
Answer: First, we find the prime factors of 512:

So, \( 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)
\( = 2^9 \)
Thus, the power form of 512 is \( 2^9 \).
In simple words: To write 512 in power form, we break it down into its smallest prime factors. Since 512 can be divided by 2 nine times, its exponential form is \( 2^9 \).

Exam Tip: Use the prime factorization method to consistently find the exponential form of numbers. Always ensure all factors are prime numbers.

 

Question 3. (ii) 343
Answer: First, we find the prime factors of 343:

So, \( 343 = 7 \times 7 \times 7 \)
\( = 7^3 \)
Thus, the power form of 343 is \( 7^3 \).
In simple words: To express 343 as a power, we find its prime factors. As 343 is the result of multiplying 7 by itself three times, its exponential form becomes \( 7^3 \).

Exam Tip: For numbers that are not easily divisible by 2, 3, or 5, try other prime numbers like 7, 11, or 13. Recognizing perfect cubes (like \( 7^3 = 343 \)) is useful.

 

Question 3. (iii) 729
Answer: First, we find the prime factors of 729:

So, \( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)
\( = 3^6 \)
Thus, the power form of 729 is \( 3^6 \).
In simple words: To write 729 as a power, we use prime factorization. Since 729 can be divided by 3 six times, its exponential form is \( 3^6 \).

Exam Tip: Remember divisibility rules, such as a number being divisible by 3 if the sum of its digits is divisible by 3. This helps in factorizing larger numbers.

 

Question 3. (iv) 3125
Answer: First, we find the prime factors of 3125:

So, \( 3125 = 5 \times 5 \times 5 \times 5 \times 5 \)
\( = 5^5 \)
Thus, the power form of 3125 is \( 5^5 \).
In simple words: To write 3125 in power form, we find its prime factors. As 3125 is the result of multiplying 5 by itself five times, its exponential form becomes \( 5^5 \).

Exam Tip: Numbers ending in 0 or 5 are always divisible by 5. This makes prime factorization easier for such numbers.

 

Question 4. નીચેના દરેકમાંથી શક્ય હોય ત્યાં મોટી સંખ્યા શોધી કાઢો:

 

Question 4. (i) \( 4^3 \) અને \( 3^4 \)
Answer: To find the larger number, we evaluate each power:
\( 4^3 = 4 \times 4 \times 4 = 64 \)
\( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \)
Since \( 81 > 64 \), we can say that \( 3^4 > 4^3 \).
In simple words: We calculate the value of \( 4^3 \) by multiplying 4 three times, which is 64. Then, we calculate \( 3^4 \) by multiplying 3 four times, which is 81. Comparing 64 and 81, we see that 81 is larger, so \( 3^4 \) is the greater number.

Exam Tip: When comparing powers, always calculate their actual values unless you are using specific exponent rules (like comparing powers with the same base or same exponent). Do not assume a larger base or exponent automatically means a larger value.

 

Question 4. (ii) \( 5^3 \) અને \( 3^5 \)
Answer: To find the larger number, we evaluate each power:
\( 5^3 = 5 \times 5 \times 5 = 125 \)
\( 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \)
Since \( 243 > 125 \), we can say that \( 3^5 > 5^3 \).
In simple words: First, we find the value of \( 5^3 \) by multiplying 5 three times, which is 125. Next, we find the value of \( 3^5 \) by multiplying 3 five times, which is 243. Comparing 125 and 243, 243 is bigger, so \( 3^5 \) is the larger number.

Exam Tip: Be careful with calculations for higher powers. It's often helpful to break down the multiplication into smaller, manageable steps (e.g., \( 3^5 = 3^2 \times 3^3 = 9 \times 27 \)).

 

Question 4. (iii) \( 2^8 \) અને \( 8^2 \)
Answer: To find the larger number, we evaluate each power:
\( 2^8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 256 \)
\( 8^2 = 8 \times 8 = 64 \)
Since \( 256 > 64 \), we can say that \( 2^8 > 8^2 \).
In simple words: We calculate \( 2^8 \) by multiplying 2 by itself eight times, getting 256. Then, we calculate \( 8^2 \) by multiplying 8 by itself two times, getting 64. As 256 is much greater than 64, \( 2^8 \) is the bigger number.

Exam Tip: Do not confuse \( 2^8 \) with \( 8^2 \). The base and exponent are not interchangeable. Calculating both values is the most reliable comparison method.

 

Question 4. (iv) \( 100^2 \) અને \( 2^{100} \)
Answer: To find the larger number, we evaluate and compare:
\( 100^2 = 100 \times 100 = 10,000 \)
For \( 2^{100} \), let's consider a smaller but significant part: \( 2^{10} = 1024 \).
So, \( 2^{100} = (2^{10})^{10} = (1024)^{10} \)
Since \( 1024 \) is already greater than \( 100 \), \( (1024)^{10} \) will be significantly larger than \( 10,000 \).
Specifically, \( (1024)^{10} > 1000^{10} = (10^3)^{10} = 10^{30} \)
Clearly, \( 10^{30} \) is much larger than \( 10,000 \).
Therefore, \( 2^{100} > 100^2 \).
In simple words: First, \( 100^2 \) means \( 100 \times 100 \), which equals 10,000. For \( 2^{100} \), we know that \( 2^{10} \) is 1024. If we take 1024 and multiply it by itself ten times, the number becomes extremely large, much bigger than 10,000. So, \( 2^{100} \) is the much larger number.

Exam Tip: For very large powers, comparing directly might not be feasible. Instead, use properties of exponents to estimate or compare. For instance, \( 2^{10} \) is a useful benchmark to remember.

 

Question 4. (v) \( 2^{10} \) અને \( 10^2 \)
Answer: To find the larger number, we evaluate each power:
\( 2^{10} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024 \)
\( 10^2 = 10 \times 10 = 100 \)
Since \( 1024 > 100 \), we can say that \( 2^{10} > 10^2 \).
In simple words: We calculate \( 2^{10} \) by multiplying 2 by itself ten times, which gives 1024. Then, we calculate \( 10^2 \) by multiplying 10 by itself two times, which gives 100. Comparing these values, 1024 is larger, so \( 2^{10} \) is the greater number.

Exam Tip: Always perform the multiplications carefully. Remember that \( 2^{10} \) is a power of two that often appears in computing, making it a good value to remember.

 

Question 5. નીચેના દરેકના અવિભાજ્ય અવયવ પાડીને તેના ગુણાકારને ઘાત સ્વરૂપે દર્શાવો:

 

Question 5. (i) 648
Answer: First, we find the prime factors of 648:

So, \( 648 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \)
\( = 2^3 \times 3^4 \)
Thus, the prime factorization of 648 in exponential form is \( 2^3 \times 3^4 \).
In simple words: To write 648 using prime factors in exponential form, we divide it by prime numbers until we reach 1. We find that 2 is a factor three times and 3 is a factor four times. So, the answer is \( 2^3 \times 3^4 \).

Exam Tip: Always divide by the smallest prime factor first to maintain consistency and reduce errors in prime factorization. Double-check your multiplication at the end.

 

Question 5. (ii) 405
Answer: First, we find the prime factors of 405:

So, \( 405 = 3 \times 3 \times 3 \times 3 \times 5 \)
\( = 3^4 \times 5 \)
Thus, the prime factorization of 405 in exponential form is \( 3^4 \times 5 \).
In simple words: To write 405 as a product of prime factors in exponential form, we divide it by prime numbers until 1 remains. We see that 3 is a factor four times and 5 is a factor once. So, the answer is \( 3^4 \times 5 \).

Exam Tip: If the number is large, consider the sum of its digits to check divisibility by 3. For 405, \( 4+0+5=9 \), which is divisible by 3, so 405 is also divisible by 3.

 

Question 5. (iii) 540
Answer: First, we find the prime factors of 540:

So, \( 540 = 2 \times 2 \times 3 \times 3 \times 3 \times 5 \)
\( = 2^2 \times 3^3 \times 5 \)
Thus, the prime factorization of 540 in exponential form is \( 2^2 \times 3^3 \times 5 \).
In simple words: To write 540 as a product of prime factors in exponential form, we divide by prime numbers. We find that 2 is a factor twice, 3 is a factor three times, and 5 is a factor once. So, the answer is \( 2^2 \times 3^3 \times 5 \).

Exam Tip: When doing prime factorization, arrange the factors in ascending order of their bases for neatness and standard mathematical notation (e.g., \( 2^x \times 3^y \times 5^z \)).

 

Question 5. (iv) 3600
Answer: First, we find the prime factors of 3600:

So, \( 3600 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \)
\( = 2^4 \times 3^2 \times 5^2 \)
Thus, the prime factorization of 3600 in exponential form is \( 2^4 \times 3^2 \times 5^2 \).
In simple words: To write 3600 as a product of prime factors in exponential form, we divide it by prime numbers until we reach 1. We find that 2 is a factor four times, 3 is a factor two times, and 5 is a factor two times. So, the answer is \( 2^4 \times 3^2 \times 5^2 \).

Exam Tip: For numbers ending in multiple zeros, you can quickly factor out powers of 10 (\( 10 = 2 \times 5 \)). For example, \( 3600 = 36 \times 100 = (2^2 \times 3^2) \times (2^2 \times 5^2) = 2^4 \times 3^2 \times 5^2 \).

 

Question 6. સાદુ રૂપ આપો:

 

Question 6. (i) \( 2 \times 10^3 \)
Answer: \( 2 \times 10^3 = 2 \times (10 \times 10 \times 10) \)
\( = 2 \times 1000 \)
\( = 2000 \)
In simple words: To simplify, first calculate \( 10^3 \) which means \( 10 \times 10 \times 10 = 1000 \). Then, multiply this 1000 by 2, which gives 2000.

Exam Tip: When multiplying by powers of 10, simply add the number of zeros indicated by the exponent to the original number. For example, \( 2 \times 10^3 \) means 2 followed by three zeros.

 

Question 6. (ii) \( 7^2 \times 2^2 \)
Answer: \( 7^2 \times 2^2 = (7 \times 7) \times (2 \times 2) \)
\( = 49 \times 4 \)
\( = 196 \)
In simple words: To simplify, calculate \( 7^2 \) which is \( 7 \times 7 = 49 \). Then calculate \( 2^2 \) which is \( 2 \times 2 = 4 \). Finally, multiply these two results: \( 49 \times 4 = 196 \).

Exam Tip: When different bases have the same exponent, you can multiply the bases first and then apply the exponent: \( (a^m \times b^m) = (a \times b)^m \). So, \( 7^2 \times 2^2 = (7 \times 2)^2 = 14^2 = 196 \).

 

Question 6. (iii) \( 2^3 \times 5 \)
Answer: \( 2^3 \times 5 = (2 \times 2 \times 2) \times 5 \)
\( = 8 \times 5 \)
\( = 40 \)
In simple words: To simplify, first calculate \( 2^3 \) which means \( 2 \times 2 \times 2 = 8 \). Then, multiply this result by 5, which gives \( 8 \times 5 = 40 \).

Exam Tip: Always evaluate the exponential term first before performing other multiplications, following the order of operations (BODMAS/PEMDAS).

 

Question 6. (iv) \( 3 \times 4^4 \)
Answer: \( 3 \times 4^4 = 3 \times (4 \times 4 \times 4 \times 4) \)
\( = 3 \times 256 \)
\( = 768 \)
In simple words: To simplify, first calculate \( 4^4 \) which means \( 4 \times 4 \times 4 \times 4 = 256 \). Then, multiply this result by 3, which gives \( 3 \times 256 = 768 \).

Exam Tip: As with other expressions, solve the power part first. Remember that \( 4^4 = (4^2)^2 = 16^2 = 256 \) for quicker calculation.

 

Question 6. (v) \( 0 \times 10^2 \)
Answer: \( 0 \times 10^2 = 0 \times (10 \times 10) \)
\( = 0 \times 100 \)
\( = 0 \)
In simple words: To simplify, first calculate \( 10^2 \) which is \( 10 \times 10 = 100 \). Then, multiply this result by 0. Any number multiplied by 0 always results in 0.

Exam Tip: A crucial property of multiplication is that multiplying any number, no matter how large, by zero always results in zero. The order of operations still applies, but the final multiplication by zero overrides other calculations.

 

Question 6. (vi) \( 5^2 \times 3^3 \)
Answer: \( 5^2 \times 3^3 = (5 \times 5) \times (3 \times 3 \times 3) \)
\( = 25 \times 27 \)
\( = 675 \)
In simple words: To simplify, first calculate \( 5^2 \) which means \( 5 \times 5 = 25 \). Then calculate \( 3^3 \) which means \( 3 \times 3 \times 3 = 27 \). Finally, multiply these two results: \( 25 \times 27 = 675 \).

Exam Tip: When multiplying numbers like 25 by another number, consider mental math strategies. For example, \( 25 \times 27 = 25 \times (20 + 7) = 25 \times 20 + 25 \times 7 = 500 + 175 = 675 \).

 

Question 6. (vii) \( 2^4 \times 3^2 \)
Answer: \( 2^4 \times 3^2 = (2 \times 2 \times 2 \times 2) \times (3 \times 3) \)
\( = 16 \times 9 \)
\( = 144 \)
In simple words: To simplify, first calculate \( 2^4 \) which means \( 2 \times 2 \times 2 \times 2 = 16 \). Then calculate \( 3^2 \) which means \( 3 \times 3 = 9 \). Finally, multiply these two results: \( 16 \times 9 = 144 \).

Exam Tip: Recognizing common squares and powers (like \( 2^4=16 \) and \( 3^2=9 \)) can speed up calculations. Also, remember that \( 12^2 = 144 \), which is useful here since \( 16 \times 9 = (4 \times 4) \times (3 \times 3) = (4 \times 3) \times (4 \times 3) = 12 \times 12 \).

 

Question 6. (viii) \( 3^2 \times 10^4 \)
Answer: \( 3^2 \times 10^4 = (3 \times 3) \times (10 \times 10 \times 10 \times 10) \)
\( = 9 \times 10,000 \)
\( = 90,000 \)
In simple words: To simplify, first calculate \( 3^2 \) which is \( 3 \times 3 = 9 \). Then calculate \( 10^4 \) which is 10,000. Finally, multiply these two results: \( 9 \times 10,000 = 90,000 \).

Exam Tip: When multiplying by powers of 10, simply append the number of zeros indicated by the exponent to the other number. Here, append four zeros to 9 to get 90,000.

 

Question 7. સાદુ રૂપ આપો:

 

Question 7. (i) \( (-4)^3 \)
Answer: \( (-4)^3 = (-4) \times (-4) \times (-4) \)
\( = (16) \times (-4) \)
\( = -64 \)
In simple words: To simplify, multiply -4 by itself three times. First, \( -4 \times -4 \) equals positive 16. Then, multiply 16 by -4, which results in -64.

Exam Tip: Remember the rule for multiplying negative numbers: an odd number of negative factors results in a negative product, while an even number of negative factors results in a positive product.

 

Question 7. (ii) \( (-3) \times (-2)^3 \)
Answer: \( (-3) \times (-2)^3 = (-3) \times (-2) \times (-2) \times (-2) \)
\( = (-3) \times (-8) \)
\( = 24 \)
In simple words: To simplify, first calculate \( (-2)^3 \) which means \( -2 \times -2 \times -2 = -8 \). Then, multiply this -8 by -3. Since a negative number multiplied by a negative number gives a positive result, \( -3 \times -8 \) is 24.

Exam Tip: Always evaluate the exponential part first, especially with negative bases. Pay close attention to the number of negative signs to determine the final sign of the product.

 

Question 7. (iii) \( (-3)^2 \times (-5)^2 \)
Answer: \( (-3)^2 \times (-5)^2 = ((-3) \times (-3)) \times ((-5) \times (-5)) \)
\( = (9) \times (25) \)
\( = 225 \)
In simple words: To simplify, calculate \( (-3)^2 \) which means \( -3 \times -3 = 9 \). Then calculate \( (-5)^2 \) which means \( -5 \times -5 = 25 \). Finally, multiply these two results: \( 9 \times 25 = 225 \).

Exam Tip: Squaring a negative number always results in a positive number because you are multiplying an even number of negative signs together.

 

Question 7. (iv) \( (-2)^3 \times (-10)^3 \)
Answer: \( (-2)^3 \times (-10)^3 = ((-2) \times (-2) \times (-2)) \times ((-10) \times (-10) \times (-10)) \)
\( = (4) \times (-2) \times (100) \times (-10) \)
\( = (-8) \times (-1000) \)
\( = 8000 \)
In simple words: To simplify, calculate \( (-2)^3 \) which means \( -2 \times -2 \times -2 = -8 \). Then calculate \( (-10)^3 \) which means \( -10 \times -10 \times -10 = -1000 \). Finally, multiply these two results: \( -8 \times -1000 \). Since a negative times a negative is positive, the answer is 8000.

Exam Tip: When both bases are negative and have the same odd exponent, the product will be positive. This is because (negative x negative x negative) x (negative x negative x negative) = (negative) x (negative) = positive.

 

Question 8. નીચેની સંખ્યાઓની સરખામણી કરોઃ

 

Question 8. (i) \( 2.7 \times 10^{12}, 1.5 \times 10^8 \)
Answer: We need to compare \( 2.7 \times 10^{12} \) and \( 1.5 \times 10^8 \).
First, let's write them in standard form:
\( 2.7 \times 10^{12} = 27 \times 10^{11} \)
\( = 2,700,000,000,000 \)

\( 1.5 \times 10^8 = 15 \times 10^7 \)
\( = 150,000,000 \)

Comparing the magnitudes, \( 2,700,000,000,000 \) has 13 digits while \( 150,000,000 \) has 9 digits. Numbers with more digits are generally larger.
So, \( 2.7 \times 10^{12} > 1.5 \times 10^8 \).
In simple words: To compare these numbers, we look at the power of 10. \( 2.7 \times 10^{12} \) means 2.7 multiplied by 1 followed by 12 zeros, resulting in a number with 13 digits. \( 1.5 \times 10^8 \) means 1.5 multiplied by 1 followed by 8 zeros, resulting in a number with 9 digits. A number with more digits is always larger, so \( 2.7 \times 10^{12} \) is greater.

Exam Tip: When comparing numbers in scientific notation, first compare the exponents of 10. The number with the larger positive exponent of 10 is usually the greater number. If exponents are equal, then compare the decimal parts.

 

Question 8. (ii) \( 4 \times 10^{14}, 3 \times 10^{17} \)
Answer: We need to compare \( 4 \times 10^{14} \) and \( 3 \times 10^{17} \).
\( 4 \times 10^{14} \) will be a number with \( 14 + 1 = 15 \) digits (the '4' plus 14 zeros).
\( 3 \times 10^{17} \) will be a number with \( 17 + 1 = 18 \) digits (the '3' plus 17 zeros).
Since \( 3 \times 10^{17} \) has more digits (18 digits) than \( 4 \times 10^{14} \) (15 digits), it is the larger number.
Therefore, \( 4 \times 10^{14} < 3 \times 10^{17} \).
In simple words: To compare these, we look at how many digits each number will have. \( 4 \times 10^{14} \) means a 4 followed by 14 zeros, making it a 15-digit number. \( 3 \times 10^{17} \) means a 3 followed by 17 zeros, making it an 18-digit number. Since 18 digits is more than 15 digits, \( 3 \times 10^{17} \) is the bigger number.

Exam Tip: When comparing numbers in scientific notation, if the exponents of 10 are different, the number with the larger exponent will always be greater, assuming both base numbers (like 4 and 3) are positive.