GSEB Class 7 Maths Solutions Chapter 13 ઘાત અને ઘાતાંક Exercise 13.2

1. ધાતાંકના નિયમોનો ઉપયોગ કરી સાદું રૂપ આપો અને જવાબને ધાત સ્વરૂપે લખોઃ

 

Question (i) \( 3^2 \times 3^4 \times 3^8 \)
Answer: \( 3^2 \times 3^4 \times 3^8 \)
\( = 3^{2+4+8} \) (કારણ કે: \( a^m \times a^n \times a^p = a^{m+n+p} \))
\( = 3^{14} \)
આમ, \( 3^2 \times 3^4 \times 3^8 = 3^{14} \)
In simple words: When you multiply numbers that have the same base but different powers, you can simply add the powers together. This gives you a new power for the same base number.

Exam Tip: Remember the rule for multiplying powers with the same base: add the exponents. Ensure all bases are identical before adding exponents.

 

Question (ii) \( 6^{15} \div 6^{10} \)
Answer: \( 6^{15} \div 6^{10} \)
\( = 6^{15-10} \) (કારણ કે: \( a^m \div a^n = a^{m-n} \))
\( = 6^5 \)
આમ, \( 6^{15} \div 6^{10} = 6^5 \)
In simple words: When you divide numbers that have the same base, you just subtract the power of the bottom number from the power of the top number. The base stays the same.

Exam Tip: For division of powers with the same base, subtract the exponent in the denominator from the exponent in the numerator. Always double-check the subtraction.

 

Question (iii) \( a^3 \times a^2 \)
Answer: \( a^3 \times a^2 \)
\( = a^{3+2} \) (કારણ કે: \( a^m \times a^n = a^{m+n} \))
\( = a^5 \)
આમ, \( a^3 \times a^2 = a^5 \)
In simple words: When letters with powers are multiplied and have the same letter as the base, you can add their powers together. This makes a new power for that same letter.

Exam Tip: This is a fundamental rule in algebra: multiplying powers with the same base means adding their exponents. Ensure the variable bases are identical before combining.

 

Question (iv) \( 7^x \times 7^2 \)
Answer: \( 7^x \times 7^2 \)
\( = 7^{x+2} \) (કારણ કે: \( a^m \times a^n = a^{m+n} \))
આમ, \( 7^x \times 7^2 = 7^{x+2} \)
In simple words: When you multiply numbers with the same base but different powers, you add the powers. Even if one power is a letter, you still add it to the other power.

Exam Tip: The rule of adding exponents for multiplication applies even if one of the exponents is a variable. Just write the sum as an expression.

 

Question (v) \( (5^2)^3 \div 5^3 \)
Answer: \( (5^2)^3 \div 5^3 \)
\( = 5^{2 \times 3} \div 5^3 \) (કારણ કે: \( (a^m)^n = a^{mn} \))
\( = 5^6 \div 5^3 \)
\( = 5^{6-3} \) (કારણ કે: \( a^m \div a^n = a^{m-n} \))
\( = 5^3 \)
આમ, \( (5^2)^3 \div 5^3 = 5^3 \)
In simple words: First, when a power is raised to another power, you multiply the powers together. Then, when dividing numbers with the same base, you subtract the powers.

Exam Tip: Always tackle the "power of a power" rule first by multiplying exponents. Then apply the division rule for exponents by subtracting them, ensuring the bases are identical.

 

Question (vi) \( 2^5 \times 5^5 \)
Answer: \( 2^5 \times 5^5 \)
\( = (2 \times 5)^5 \) (કારણ કે: \( a^m \times b^m = (ab)^m \))
\( = 10^5 \)
આમ, \( 2^5 \times 5^5 = 10^5 \)
In simple words: When numbers have different bases but the same power, you can multiply the bases first and then apply the common power to the result.

Exam Tip: Remember this shortcut: if exponents are the same, multiply the bases and keep the exponent. It simplifies calculations greatly.

 

Question (vii) \( a^4 \times b^4 \)
Answer: \( a^4 \times b^4 \)
\( = (ab)^4 \) (કારણ કે: \( a^m \times b^m = (ab)^m \))
આમ, \( a^4 \times b^4 = (ab)^4 \)
In simple words: If two different letters have the same power and are multiplied together, you can multiply the letters first and then raise the whole product to that common power.

Exam Tip: This rule is useful for simplifying expressions. When different bases have identical exponents, combine the bases under a single exponent.

 

Question (viii) \( (3^4)^3 \)
Answer: \( (3^4)^3 \)
\( = 3^{4 \times 3} \) (કારણ કે: \( (a^m)^n = a^{mn} \))
\( = 3^{12} \)
આમ, \( (3^4)^3 = 3^{12} \)
In simple words: When you have a number already raised to a power, and then that whole thing is raised to another power, you just multiply the two powers together. The base stays the same.

Exam Tip: The "power of a power" rule (multiplying exponents) is crucial for simplifying complex exponential terms. Don't add them by mistake.

 

Question (ix) \( (2^{20} \div 2^{15}) \times 2^3 \)
Answer: \( (2^{20} \div 2^{15}) \times 2^3 \)
\( = (2^{20-15}) \times 2^3 \) (કારણ કે: \( a^m \div a^n = a^{m-n} \))
\( = 2^5 \times 2^3 \)
\( = 2^{5+3} \) (કારણ કે: \( a^m \times a^n = a^{m+n} \))
\( = 2^8 \)
આમ, \( (2^{20} \div 2^{15}) \times 2^3 = 2^8 \)
In simple words: First, handle the division: subtract the powers because the bases are the same. Then, for the multiplication, add the resulting power to the next power, again keeping the base the same.

Exam Tip: Follow the order of operations (PEMDAS/BODMAS): parentheses first. Apply division rule (subtract exponents), then multiplication rule (add exponents).

 

Question (x) \( 8^t \div 8^2 \)
Answer: \( 8^t \div 8^2 \)
\( = 8^{t-2} \) (કારણ કે: \( a^m \div a^n = a^{m-n} \))
આમ, \( 8^t \div 8^2 = 8^{t-2} \)
In simple words: When you divide numbers with the same base, you subtract the powers. Even if one power is a letter, you simply write it as a subtraction in the new power.

Exam Tip: The exponent rule for division applies to variable exponents as well. Ensure the base remains unchanged, and the exponents are correctly subtracted.

2. સાદું રૂપ આપી નીચેના દરેકને ઘાત સ્વરૂપે દર્શાવોઃ

 

Question (i) \( \frac{2^{3} \times 3^{4} \times 4}{3 \times 32} \)
Answer: \( \frac{2^{3} \times 3^{4} \times 4}{3 \times 32} \)
\( = \frac{2^{3} \times 3^{4} \times 2^{2}}{3 \times 2^{5}} \)
\( = \frac{2^{3+2} \times 3^{4}}{3^1 \times 2^{5}} \)
\( = \frac{2^{5} \times 3^{4}}{2^{5} \times 3^1} \)
\( = 2^{5-5} \times 3^{4-1} \)
\( = 2^0 \times 3^3 \)
\( = 1 \times 3^3 \)
\( = 3^3 \)
In simple words: First, change all numbers to their prime factors with powers. Then, group the bases that are the same. When dividing with the same base, subtract the bottom power from the top power. Any number to the power of zero is 1.

Exam Tip: Always convert composite numbers (like 4 and 32) into their prime factor bases (2^2 and 2^5) to simplify the expression using exponent rules. Simplify powers of zero to one.

 

Question (ii) \( [(5^2)^3 \times 5^4] \div 5^7 \)
Answer: \( [(5^2)^3 \times 5^4] \div 5^7 \)
\( = [5^{2 \times 3} \times 5^4] \div 5^7 \)
\( = [5^6 \times 5^4] \div 5^7 \)
\( = [5^{6+4}] \div 5^7 \)
\( = 5^{10} \div 5^7 \)
\( = 5^{10-7} \)
\( = 5^3 \)
In simple words: First, multiply the exponents inside the bracket. Then, add the exponents for the numbers being multiplied. Finally, subtract the exponents for the division part.

Exam Tip: Remember the order of operations: Parentheses (or brackets) first. Inside the brackets, apply the "power of a power" rule, then the multiplication rule. Finally, apply the division rule.

 

Question (iii) \( 25^4 \div 5^3 \)
Answer: \( 25^4 \div 5^3 \)
\( = (5^2)^4 \div 5^3 \)
\( = 5^{2 \times 4} \div 5^3 \)
\( = 5^8 \div 5^3 \)
\( = 5^{8-3} \)
\( = 5^5 \)
In simple words: Change the base of the first number to match the second number. So, 25 becomes 5 squared. Then, multiply the powers. After that, divide by subtracting the powers.

Exam Tip: Always try to make the bases of all numbers the same. This allows you to apply the exponent rules for multiplication and division easily. Recognize composite numbers as powers of a prime base.

 

Question (iv) \( \frac{3 \times 7^{2} \times 11^{8}}{21 \times 11^{3}} \)
Answer: \( \frac{3 \times 7^{2} \times 11^{8}}{21 \times 11^{3}} \)
\( = \frac{3 \times 7^{2} \times 11^{8}}{3 \times 7 \times 11^{3}} \)
\( = \frac{3}{3} \times \frac{7^{2}}{7} \times \frac{11^{8}}{11^{3}} \)
\( = 3^{1-1} \times 7^{2-1} \times 11^{8-3} \)
\( = 3^0 \times 7^1 \times 11^5 \)
\( = 1 \times 7 \times 11^5 \)
\( = 7 \times 11^5 \)
In simple words: First, break down any composite number in the denominator (like 21) into its prime factors. Then, group fractions with the same base and subtract their powers. Simplify any base raised to the power of zero to one.

Exam Tip: Prime factorization is key to simplifying complex fractions with exponents. Cancel out common factors or apply exponent rules for division to simplify terms with the same base.

 

Question (v) \( \frac{3^{7}}{3^{4} \times 3^{3}} \)
Answer: \( \frac{3^{7}}{3^{4} \times 3^{3}} \)
\( = \frac{3^{7}}{3^{4+3}} \)
\( = \frac{3^{7}}{3^{7}} \)
\( = 3^{7-7} \)
\( = 3^0 \)
\( = 1 \)
In simple words: First, add the powers of the numbers in the denominator because they have the same base and are multiplied. Then, divide the numbers by subtracting the power of the denominator from the power of the numerator. Any number to the power of zero is one.

Exam Tip: When simplifying fractions with exponents, combine terms in the numerator and denominator separately before performing division. Remember that anything raised to the power of zero equals one.

 

Question (vi) \( 2^0 + 3^0 + 4^0 \)
Answer: \( 2^0 + 3^0 + 4^0 \)
\( = 1 + 1 + 1 \) (કારણ કે: \( a^0 = 1 \))
\( = 3 \)
In simple words: Any number (except zero itself) raised to the power of zero is always one. So, just add all the ones together.

Exam Tip: A common trap is to assume 0^0 is 1. Remind yourself that any non-zero number raised to the power of zero is 1. This is a quick and easy calculation.

 

Question (vii) \( 2^0 \times 3^0 \times 4^0 \)
Answer: \( 2^0 \times 3^0 \times 4^0 \)
\( = 1 \times 1 \times 1 \) (કારણ કે: \( a^0 = 1 \))
\( = 1 \)
In simple words: Since any number (not zero) to the power of zero is one, you are just multiplying one by one by one. The result is always one.

Exam Tip: This is a straightforward application of the zero exponent rule. All terms become 1, so the product is simply 1.

 

Question (viii) \( (3^0 + 2^0) \times 5^0 \)
Answer: \( (3^0 + 2^0) \times 5^0 \)
\( = (1 + 1) \times 1 \) (કારણ કે: \( a^0 = 1 \))
\( = 2 \times 1 \)
\( = 2 \)
In simple words: First, remember that any number (not zero) to the power of zero is one. So, replace each term with one, do the addition inside the brackets, and then multiply.

Exam Tip: Apply the zero exponent rule first to simplify each term, then follow the order of operations (parentheses before multiplication).

 

Question (ix) \( \frac{2^{8} \times a^{5}}{4^{3} \times a^{3}} \)
Answer: \( \frac{2^{8} \times a^{5}}{4^{3} \times a^{3}} \)
\( = \frac{2^{8} \times a^{5}}{(2^{2})^{3} \times a^{3}} \)
\( = \frac{2^{8} \times a^{5}}{2^{6} \times a^{3}} \)
\( = 2^{8-6} \times a^{5-3} \)
\( = 2^2 \times a^2 \)
\( = (2a)^2 \)
In simple words: First, change the number 4 into two squared. Then, multiply the powers. After that, divide the parts that have the same base by subtracting the powers. Finally, combine the result as a single term raised to a power.

Exam Tip: Convert composite bases (like 4 to 2^2) to prime bases to enable application of exponent rules. Always simplify powers of powers before combining terms.

 

Question (x) \( (\frac{a^{5}}{a^{3}}) \times a^{8} \)
Answer: \( (\frac{a^{5}}{a^{3}}) \times a^{8} \)
\( = a^{5-3} \times a^{8} \)
\( = a^2 \times a^{8} \)
\( = a^{2+8} \)
\( = a^{10} \)
In simple words: First, divide the terms inside the parentheses by subtracting their powers. Then, multiply the result by the next term, which means adding their powers.

Exam Tip: Process the division within the parentheses first by subtracting exponents, then multiply the result by adding exponents. Remember the order of operations.

 

Question (xi) \( \frac{4^{5} \times a^{8} b^{3}}{4^{5} \times a^{5} b^{2}} \)
Answer: \( \frac{4^{5} \times a^{8} b^{3}}{4^{5} \times a^{5} b^{2}} \)
\( = \frac{4^{5}}{4^{5}} \times \frac{a^{8}}{a^{5}} \times \frac{b^{3}}{b^{2}} \)
\( = 4^{5-5} \times a^{8-5} \times b^{3-2} \)
\( = 4^0 \times a^3 \times b^1 \)
\( = 1 \times a^3 \times b \)
\( = a^3b \)
In simple words: Separate the expression into parts that have the same base. For each part, subtract the power of the bottom number from the power of the top number. Simplify any term raised to the power of zero to one.

Exam Tip: Break down complex fractions into simpler ones with common bases. This allows you to apply the division rule of exponents to each variable or number independently, then simplify.

 

Question (xii) \( (2^3 \times 2)^2 \)
Answer: \( (2^3 \times 2)^2 \)
\( = (2^{3+1})^2 \)
\( = (2^4)^2 \)
\( = 2^{4 \times 2} \)
\( = 2^8 \)
In simple words: First, add the powers of the numbers inside the brackets because they have the same base and are multiplied. Then, multiply the resulting power by the power outside the bracket.

Exam Tip: Simplify inside the parentheses first by applying the multiplication rule for exponents (adding exponents). Then, apply the "power of a power" rule (multiplying exponents).

3. ખરાં છે કે ખોટાં તે કહો અને તમારા જવાબને ચકાસોઃ

 

Question (i) \( 10 \times 10^{11} = 100^{11} \)
Answer: ડાબા. (LHS) \( = 10 \times 10^{11} \)
\( = 10^1 \times 10^{11} \)
\( = 10^{1+11} \)
\( = 10^{12} \)
જ.બા. (RHS) \( = 100^{11} \)
\( = (10^2)^{11} \)
\( = 10^{2 \times 11} \)
\( = 10^{22} \)
અહીં ડાબા. \( \neq \) જ.બા.
\( \implies \) આપેલું વિધાન ખોટું છે.
In simple words: Calculate both sides of the equation. On the left, add the powers. On the right, multiply the powers. Since the results are different, the statement is false.

Exam Tip: For true/false questions involving exponents, always simplify both sides of the equation separately to their simplest exponential form with the same base, then compare them.

 

Question (ii) \( 2^3 > 5^2 \)
Answer: \( 2^3 = 2 \times 2 \times 2 = 8 \)
\( 5^2 = 5 \times 5 = 25 \)
\( 8 < 25 \)
\( \implies 2^3 < 5^2 \)
\( \implies \) આપેલું વિધાન ખોટું છે.
In simple words: Work out the value of each number with its power. Then, compare the two results to see if the greater than sign is correct. If it's not, the statement is false.

Exam Tip: Calculate the exact numerical values of exponential terms when comparing magnitudes. Do not try to compare exponents directly unless the bases are identical.

 

Question (iii) \( 2^3 \times 3^2 = 6^5 \)
Answer: ડાબા. (LHS) \( = 2^3 \times 3^2 \)
\( = (2 \times 2 \times 2) \times (3 \times 3) \)
\( = 8 \times 9 \)
\( = 72 \)
જ.બા. (RHS) \( = 6^5 \)
\( = 6 \times 6 \times 6 \times 6 \times 6 \)
\( = 7776 \)
હવે, ડાબા. \( \neq \) જ.બા.
\( \implies \) આપેલું વિધાન ખોટું છે.
In simple words: Figure out the value of the left side by doing the multiplication of powers. Then, figure out the value of the right side. Since they are not the same, the statement is false.

Exam Tip: Do not combine bases that are different (2 and 3) by multiplying them and adding exponents. Calculate each term separately and then perform multiplication.

 

Question (iv) \( 3^0 = (1000)^0 \)
Answer: ડાબા. (LHS) \( = 3^0 = 1 \) (કારણ કે: \( a^0 = 1 \))
જ.બા. (RHS) \( = (1000)^0 = 1 \) (કારણ કે: \( a^0 = 1 \))
\( \implies \) ડાબા. \( = \) જ.બા.
\( \implies \) આપેલું વિધાન સાચું છે.
In simple words: Remember that any non-zero number raised to the power of zero is always one. Since both sides become one, they are equal, and the statement is true.

Exam Tip: This question tests the fundamental rule of exponents: any non-zero number raised to the power of zero is 1. This makes the comparison very simple.

4. નીચેના ગુણાકારના અવિભાજ્ય અવયવ પાડી તેને ધાત સ્વરૂપે દર્શાવો:

 

Question (i) \( 108 \times 192 \)
Answer:

\( 108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3 \)
\( 192 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^6 \times 3^1 \)
\( \implies 108 \times 192 = (2^2 \times 3^3) \times (2^6 \times 3^1) \)
\( = 2^2 \times 2^6 \times 3^3 \times 3^1 \)
\( = 2^{2+6} \times 3^{3+1} \)
\( = 2^8 \times 3^4 \)
આમ, \( 108 \times 192 = 2^8 \times 3^4 \)
In simple words: First, find the prime factors for both 108 and 192 separately and write them using powers. Then, multiply these powered forms together. Group the same bases and add their powers to get the final answer.

Exam Tip: Prime factorization is a key skill here. Organize your factorization neatly. When combining, group terms with the same base and apply the multiplication rule for exponents (adding powers).

 

Question (ii) \( 270 \)
Answer:

\( 270 = 2 \times 3 \times 3 \times 3 \times 5 \)
\( = 2^1 \times 3^3 \times 5^1 \)
\( = 2 \times 3^3 \times 5 \)
આમ, \( 270 = 2 \times 3^3 \times 5 \)
In simple words: Break down the number 270 into its prime factors. Write down each prime factor and count how many times it appears, then express it using powers.

Exam Tip: Practice prime factorization using a division method. Ensure you only use prime numbers as divisors and continue until you reach 1. This converts the number into its exponential form.

 

Question (iii) \( 729 \times 64 \)
Answer:

\( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \)
\( 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 \)
\( \implies 729 \times 64 = (3^6) \times (2^6) \)
\( = (3 \times 2)^6 \)
\( = 6^6 \)
આમ, \( 729 \times 64 = 6^6 \)
In simple words: First, find the prime factors for both 729 and 64 and write them using powers. Then, since both numbers have the same power, multiply their bases and keep the common power.

Exam Tip: When two numbers have different bases but the same exponent, their product can be written as the product of their bases raised to that common exponent. This simplifies the expression greatly.

 

Question (iv) \( 768 \)
Answer:

\( 768 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \)
\( = 2^8 \times 3^1 \)
\( = 2^8 \times 3 \)
આમ, \( 768 = 2^8 \times 3 \)
In simple words: Break down the number 768 into its prime factors using repeated division. Count how many times each prime factor appears and write the number in exponential form.

Exam Tip: Be systematic when performing prime factorization. Use a factor tree or repeated division, making sure to list all prime factors and count their occurrences accurately to determine the exponents.

5. સાદું રૂપ આપોઃ

 

Question (i) \( \frac{(2^{5})^{2} \times 7^{3}}{8^{3} \times 7} \)
Answer: \( \frac{(2^{5})^{2} \times 7^{3}}{8^{3} \times 7} \)
\( = \frac{2^{5 \times 2} \times 7^{3}}{(2^{3})^{3} \times 7^1} \)
\( = \frac{2^{10} \times 7^{3}}{2^{9} \times 7^1} \)
\( = 2^{10-9} \times 7^{3-1} \)
\( = 2^1 \times 7^2 \)
\( = 2 \times 49 \)
\( = 98 \)
In simple words: First, express 8 as two cubed. Then, multiply the exponents where a power is raised to another power. After that, subtract the exponents for terms with the same base that are being divided. Finally, calculate the numerical result.

Exam Tip: Simplify the bases first (e.g., 8 to 2^3). Apply the "power of a power" rule (multiply exponents) and then the division rule (subtract exponents) for each distinct base before performing final multiplication.

 

Question (ii) \( \frac{25 \times 5^{2} \times t^{8}}{10^{3} \times t^{4}} \)
Answer: \( \frac{25 \times 5^{2} \times t^{8}}{10^{3} \times t^{4}} \)
\( = \frac{5^{2} \times 5^{2} \times t^{8}}{(2 \times 5)^{3} \times t^{4}} \)
\( = \frac{5^{2+2} \times t^{8}}{2^{3} \times 5^{3} \times t^{4}} \)
\( = \frac{5^{4} \times t^{8}}{2^{3} \times 5^{3} \times t^{4}} \)
\( = \frac{1}{2^{3}} \times 5^{4-3} \times t^{8-4} \)
\( = \frac{1}{2^3} \times 5^1 \times t^4 \)
\( = \frac{5 t^4}{8} \)
In simple words: Convert 25 to 5 squared and 10 to two times five. Then, add the powers of similar bases in the numerator. Next, apply the power to both numbers in the denominator. Finally, subtract the powers for terms with the same base that are being divided.

Exam Tip: Always convert composite numbers (like 25 and 10) to their prime factor forms first. This enables consistent application of exponent rules for all parts of the expression, whether numbers or variables.

 

Question (iii) \( \frac{3^{5} \times 10^{5} \times 25}{5^{7} \times 6^{5}} \)
Answer: \( \frac{3^{5} \times 10^{5} \times 25}{5^{7} \times 6^{5}} \)
\( = \frac{3^{5} \times (2 \times 5)^{5} \times 5^{2}}{5^{7} \times (2 \times 3)^{5}} \)
\( = \frac{3^{5} \times 2^{5} \times 5^{5} \times 5^{2}}{5^{7} \times 2^{5} \times 3^{5}} \)
\( = \frac{2^{5} \times 3^{5} \times 5^{5+2}}{2^{5} \times 3^{5} \times 5^{7}} \)
\( = \frac{2^{5} \times 3^{5} \times 5^{7}}{2^{5} \times 3^{5} \times 5^{7}} \)
\( = 2^{5-5} \times 3^{5-5} \times 5^{7-7} \)
\( = 2^0 \times 3^0 \times 5^0 \)
\( = 1 \times 1 \times 1 \)
\( = 1 \)
In simple words: First, break down composite bases (like 10, 25, and 6) into their prime factors. Distribute the powers. Then, group terms with the same base and add their powers in the numerator and denominator separately. Finally, subtract the powers for division. All terms will cancel out, leaving one.

Exam Tip: Complex expressions require systematic prime factorization of all bases. Carefully apply the rules for "power of a product" and then combine like bases by adding/subtracting exponents. A result of 1 is common when all factors cancel out.