GSEB Class 7 Maths Solutions Chapter 13 Exponents and Powers Exercise 13.1

Question 1. Find the value of:
(i) \( 2^6 \)
(ii) \( 9^3 \)
(iii) \( 11^2 \)
(iv) \( 5^4 \)
Answer:
(i) We find that \( 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \). Therefore, the value of \( 2^6 \) is 64.
(ii) We see that \( 9^3 = 9 \times 9 \times 9 = 729 \). Thus, the value of \( 9^3 \) is 729.
(iii) We have \( 11^2 = 11 \times 11 = 121 \). Hence, the value of \( 11^2 \) is 121.
(iv) We can calculate \( 5^4 = 5 \times 5 \times 5 \times 5 = 625 \). So, the value of \( 5^4 \) is 625.
In simple words: To find the value, multiply the base number by itself as many times as the power indicates.

Exam Tip: Remember that the exponent (the small number) tells you how many times to multiply the base number by itself, not to multiply the base by the exponent.

Question 2. Express the following in exponential form:
(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) \( a \times a \times a \times c \times c \times c \times c \times d \)
Answer:
(i) We observe that \( 6 \times 6 \times 6 \times 6 \) can be written as \( 6^4 \).
(ii) We have \( t \times t \) which is simply \( t^2 \).
(iii) We find that \( b \times b \times b \times b \) can be shown as \( b^4 \).
(iv) We see that \( 5 \times 5 \times 7 \times 7 \times 7 \) becomes \( 5^2 \times 7^3 \).
(v) We note that \( 2 \times 2 \times a \times a \) is represented as \( 2^2 \times a^2 \).
(vi) We discover that \( a \times a \times a \times c \times c \times c \times c \times d \) simplifies to \( (a \times a \times a) \times (c \times c \times c \times c) \times d = a^3 \times c^4 \times d \).
In simple words: Count how many times each number or letter repeats, then write that count as a small number (exponent) above the base.

Exam Tip: Ensure that you group identical bases together before assigning their respective exponents. Different bases need separate exponential terms.

Question 3. Express each of the following numbers using exponential notation:
(i) 512
(ii) 343
(iii) 729
(iv) 3125
Answer:
(i) For 512:
\[ \begin{array}{c|c} 2 & 512 \\ \hline 2 & 256 \\ 2 & 128 \\ 2 & 64 \\ 2 & 32 \\ 2 & 16 \\ 2 & 8 \\ 2 & 4 \\ 2 & 2 \\ \cline{2-2} & 1 \\ \end{array} \]
We observe that \( 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9 \). Therefore, the exponential form of 512 is \( 2^9 \).
(ii) For 343:
\[ \begin{array}{c|c} 7 & 343 \\ \hline 7 & 49 \\ 7 & 7 \\ \cline{2-2} & 1 \\ \end{array} \]
We find that \( 343 = 7 \times 7 \times 7 = 7^3 \). Thus, the exponential notation of 343 is \( 7^3 \).
(iii) For 729:
\[ \begin{array}{c|c} 3 & 729 \\ \hline 3 & 243 \\ 3 & 81 \\ 3 & 27 \\ 3 & 9 \\ 3 & 3 \\ \cline{2-2} & 1 \\ \end{array} \]
We discover that \( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \). Hence, the exponential form of 729 is \( 3^6 \).
(iv) For 3125:
\[ \begin{array}{c|c} 5 & 3125 \\ \hline 5 & 625 \\ 5 & 125 \\ 5 & 25 \\ 5 & 5 \\ \cline{2-2} & 1 \\ \end{array} \]
We calculate that \( 3125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5 \). Therefore, the exponential form of 3125 is \( 5^5 \).
In simple words: To write a number in exponential notation, break it down into its prime factors. Then, count how many times each prime factor appears and use that count as its exponent.

Exam Tip: Use prime factorization (repeated division by prime numbers) to consistently break down the number until you reach 1. This method helps prevent errors.

Question 4. Identify the greater number, wherever possible, in each of the following:
(i) \( 4^3 \) or \( 3^4 \)
(ii) \( 5^3 \) or \( 3^5 \)
(iii) \( 2^8 \) or \( 8^2 \)
(iv) \( 100^2 \) or \( 2^{100} \)
(v) \( 2^{10} \) or \( 10^2 \)
Answer:
(i) We calculate \( 4^3 = 4 \times 4 \times 4 = 64 \). We also find \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \). As \( 81 > 64 \), it means that \( 3^4 \) is greater than \( 4^3 \).
(ii) We compute \( 5^3 = 5 \times 5 \times 5 = 125 \). We also get \( 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \). Because \( 243 > 125 \), it shows that \( 3^5 \) is larger than \( 5^3 \).
(iii) We determine \( 2^8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 256 \). We also find \( 8^2 = 8 \times 8 = 64 \). Given \( 256 > 64 \), this indicates \( 2^8 \) is greater than \( 8^2 \).
(iv) We calculate \( 100^2 = 100 \times 100 = 10000 \). For \( 2^{100} \), we can write it as \( (2^{10})^{10} \), which means \( (1024)^{10} \). This can be further expressed as \( (1024 \times 1024)^5 \), resulting in \( (1048576)^5 \). Since \( 1048576 \) is much larger than \( 10000 \), it clearly follows that \( (1048576)^5 \) is far greater than \( 100^2 \). Thus, \( 2^{100} \) is the greater number.
(v) Given \( 2^{10} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024 \) and \( 10^2 = 10 \times 10 = 100 \). As \( 1024 \) is greater than \( 100 \), this implies that \( 2^{10} \) is the larger number.
In simple words: To compare numbers in exponential form, calculate their actual values first. Then, simply see which number is bigger. For very large exponents, you can sometimes compare by how many digits they have.

Exam Tip: For larger exponents, consider using properties of exponents (e.g., \( (a^m)^n = a^{mn} \)) to simplify before comparing, or determine which number will have significantly more digits.

Question 5. Express each of the following as product of powers of their prime factors:
(i) 648
(ii) 405
(iii) 540
(iv) 3,600
Answer:
(i) For 648:
\( 648 = 2 \times 324 \)
\( = 2 \times 2 \times 162 \)
\( = 2 \times 2 \times 2 \times 81 \)
\( = 2 \times 2 \times 2 \times 3 \times 27 \)
\( = 2 \times 2 \times 2 \times 3 \times 3 \times 9 \)
\( = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \)
\( = 2^3 \times 3^4 \)
Therefore, \( 648 \) can be expressed as \( 2^3 \times 3^4 \).
(ii) For 405:
\( 405 = 3 \times 135 \)
\( = 3 \times 3 \times 45 \)
\( = 3 \times 3 \times 3 \times 15 \)
\( = 3 \times 3 \times 3 \times 3 \times 5 \)
\( = 3^4 \times 5^1 \)
Thus, \( 405 \) is written as \( 3^4 \times 5 \).
(iii) For 540:
\( 540 = 2 \times 270 \)
\( = 2 \times 2 \times 135 \)
\( = 2 \times 2 \times 3 \times 45 \)
\( = 2 \times 2 \times 3 \times 3 \times 15 \)
\( = 2 \times 2 \times 3 \times 3 \times 3 \times 5 \)
\( = 2^2 \times 3^3 \times 5 \)
Hence, \( 540 \) equals \( 2^2 \times 3^3 \times 5 \).
(iv) For 3600:
\( 3600 = 2 \times 1800 \)
\( = 2 \times 2 \times 900 \)
\( = 2 \times 2 \times 2 \times 450 \)
\( = 2 \times 2 \times 2 \times 2 \times 225 \)
\( = 2 \times 2 \times 2 \times 2 \times 3 \times 75 \)
\( = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 25 \)
\( = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \)
\( = 2^4 \times 3^2 \times 5^2 \)
Therefore, \( 3600 \) can be shown as \( 2^4 \times 3^2 \times 5^2 \).
In simple words: To show a number as powers of its prime factors, keep dividing it by the smallest prime numbers until only prime numbers are left. Then, write each prime factor with an exponent showing how many times it appeared.

Exam Tip: Always use only prime numbers (2, 3, 5, 7, 11, etc.) for factorization. Make sure you divide completely until the quotient is 1.

Question 6. Simplify:
(i) \( 2 \times 10^3 \)
(ii) \( 7^2 \times 2^2 \)
(iii) \( 2^3 \times 5 \)
(iv) \( 3 \times 4^4 \)
(v) \( 0 \times 10^2 \)
(vi) \( 5^2 \times 3^3 \)
(vii) \( 2^4 \times 3^2 \)
(viii) \( 3^2 \times 10^4 \)
Answer:
(i) First, calculate \( 10^3 \) which is \( 10 \times 10 \times 10 = 1000 \). Then, multiply this by 2: \( 2 \times 1000 = 2000 \).
(ii) We find \( 7^2 = 7 \times 7 = 49 \) and \( 2^2 = 2 \times 2 = 4 \). Multiplying these results in \( 49 \times 4 = 196 \).
(iii) First, compute \( 2^3 = 2 \times 2 \times 2 = 8 \). Then, multiply by 5: \( 8 \times 5 = 40 \).
(iv) Calculate \( 4^4 \) first, which is \( 4 \times 4 \times 4 \times 4 = 256 \). After that, multiply by 3: \( 3 \times 256 = 768 \).
(v) First, \( 10^2 = 10 \times 10 = 100 \). Then, multiply by zero: \( 0 \times 100 = 0 \). So, the final result is 0.
(vi) We compute \( 5^2 = 5 \times 5 = 25 \) and \( 3^3 = 3 \times 3 \times 3 = 27 \). Multiplying these values gives \( 25 \times 27 = 675 \).
(vii) Calculate \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \) and \( 3^2 = 3 \times 3 = 9 \). Multiplying these results in \( 16 \times 9 = 144 \).
(viii) First, compute \( 3^2 = 3 \times 3 = 9 \). Next, find \( 10^4 = 10 \times 10 \times 10 \times 10 = 10000 \). Multiplying these together yields \( 9 \times 10000 = 90000 \).
In simple words: First, work out the value of any number raised to a power. Then, do the multiplication to get the final simple answer.

Exam Tip: Remember the order of operations (PEMDAS/BODMAS) where exponents are evaluated before multiplication. Be careful with calculations, especially with larger numbers.

Question 7. Simplify:
(i) \( (-4)^3 \)
(ii) \( (-3) \times (-2)^3 \)
(iii) \( (-3)^2 \times (-5)^2 \)
(iv) \( (-2)^3 \times (-10)^3 \)
Answer:
(i) To find \( (-4)^3 \), we multiply \( (-4) \) by itself three times: \( (-4) \times (-4) \times (-4) \). This can be thought of as \( (-1)^3 \times 4 \times 4 \times 4 \), which simplifies to \( -1 \times 64 = -64 \).
(ii) First, calculate \( (-2)^3 = (-2) \times (-2) \times (-2) = -8 \). Then multiply by \( (-3) \): \( (-3) \times (-8) = 24 \).
(iii) We find \( (-3)^2 = (-3) \times (-3) = 9 \) and \( (-5)^2 = (-5) \times (-5) = 25 \). Multiplying these results gives \( 9 \times 25 = 225 \).
(iv) First, compute \( (-2)^3 = -8 \). Next, find \( (-10)^3 = -1000 \). Multiplying these two results in \( -8 \times -1000 = 8000 \).
In simple words: When simplifying expressions with negative bases and exponents, pay close attention to whether the exponent is odd or even, as this affects the sign of the result.

Exam Tip: An odd exponent on a negative base results in a negative number. An even exponent on a negative base results in a positive number.

Question 8. Compare the following numbers:
(i) \( 2.7 \times 10^{12}; 1.5 \times 10^8 \)
(ii) \( 4 \times 10^{14}; 3 \times 10^{17} \)
Answer:
(i) We can rewrite \( 2.7 \times 10^{12} \) as \( \frac{27}{10} \times 10^{12} = 27 \times 10^{11} \). Similarly, \( 1.5 \times 10^8 \) can be expressed as \( \frac{15}{10} \times 10^8 = 15 \times 10^7 \). When we expand these, \( 27 \times 10^{11} \) becomes 2,700,000,000,000 and \( 15 \times 10^7 \) becomes 150,000,000. Since 2,700,000,000,000 is much larger than 150,000,000, it clearly indicates that \( 2.7 \times 10^{12} \) is greater than \( 1.5 \times 10^8 \).
(ii) The number \( 4 \times 10^{14} \) has 15 digits in total. On the other hand, \( 3 \times 10^{17} \) contains 18 digits. A number that includes more digits is always larger. Therefore, \( 3 \times 10^{17} \) will be greater than \( 4 \times 10^{14} \).
In simple words: To compare very large numbers in scientific notation, first look at the exponent. A larger exponent usually means a bigger number. If exponents are similar, then compare the base numbers. You can also compare the total number of digits.

Exam Tip: When comparing numbers in scientific notation, the exponent of 10 is the primary factor. The number with the larger exponent is generally the greater number. If exponents are the same, then compare the decimal parts.