CBSE Class 7 Mathematics Exponents And Powers Worksheet Set 07

Mathematics Worksheet - Exponents and Powers

Question 1. Express the following in exponential form.
a. \( (-2) \times (-2) \times (-2) \times (-2) \times (-2) \)
b. \( (-x) \times (-x) \times (-x) \times y \times y \times y \times y \)
c. \( 5 \times 5 \times 5 \times t \times t \times t \times c \times c \times c \)
d. \( a \times a \times a \times a \times a \times a \times a \)
Answer:
a. Since -2 is multiplied 5 times, we write it as \( (-2)^5 \).
b. Here, -x is multiplied 3 times and y is multiplied 4 times. We write this as \( (-x)^3 \times y^4 \).
c. We can group these as 5, t, and c, each multiplied 3 times. We write it as \( 5^3 \times t^3 \times c^3 \), which is also \( (5tc)^3 \).
d. Since a is multiplied 7 times, we write it as \( a^7 \).
In simple words: To write in exponential form, count how many times a number or letter multiplies itself. Write that count on the top right.

Exam Tip: When a negative number is inside parentheses, make sure to keep the negative sign inside the parentheses in your final power form.

Question 2. Write the base and exponents in each of the following.
a. \( (3)^7 \)
b. \( \left(\frac{2}{3}\right)^4 \)
c. \( (-7)^5 \)
d. \( (-4)^6 \)
Answer:
a. Base = 3, Exponent = 7
b. Base = \( \frac{2}{3} \), Exponent = 4
c. Base = -7, Exponent = 5
d. Base = -4, Exponent = 6
In simple words: The big number at the bottom is the base. The small number on top is the exponent.

Exam Tip: If the base has a minus sign, include that minus sign as part of the base.

Question 3. Simplify.
a. \( 2^4 \)
b. \( 3^4 \)
c. \( 7^4 \)
d. \( 10^5 \)
Answer:
a. \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \)
b. \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \)
c. \( 7^4 = 7 \times 7 \times 7 \times 7 = 2401 \)
d. \( 10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000 \)
In simple words: To simplify, multiply the base by itself as many times as the exponent says.

Exam Tip: For powers of 10, the exponent tells you exactly how many zeros to write after the number 1.

Question 4. Express as product of their powers.
a. \( 729 \times 9 \)
b. \( 1080 \times 216 \)
c. \( 1200 \)
d. \( 864 \)
e. \( -1000 \)
f. \( \frac{8}{729} \)
g. \( \frac{81}{2401} \)
Answer:
a. First, find prime factors. \( 729 = 3^6 \) and \( 9 = 3^2 \).
\( \implies 3^6 \times 3^2 = 3^{6+2} = 3^8 \) (or \( 9^4 \)).
b. Prime factors for \( 1080 = 2^3 \times 3^3 \times 5 \).
Prime factors for \( 216 = 2^3 \times 3^3 \).
\( \implies (2^3 \times 3^3 \times 5) \times (2^3 \times 3^3) = 2^{3+3} \times 3^{3+3} \times 5 = 2^6 \times 3^6 \times 5 \).
c. Find the prime factors of 1200.
\( 1200 = 12 \times 100 = 2^2 \times 3 \times 2^2 \times 5^2 = 2^4 \times 3 \times 5^2 \).
d. Find the prime factors of 864.
\( 864 = 2^5 \times 3^3 \).
e. For -1000, we can write it as \( (-10)^3 \) or as prime factors: \( -(2^3 \times 5^3) \).
f. Factor the top and bottom.
\( 8 = 2^3 \) and \( 729 = 3^6 = 9^3 \).
\( \implies \frac{2^3}{3^6} \) or \( \left(\frac{2}{9}\right)^3 \).
g. Factor the top and bottom.
\( 81 = 3^4 \) and \( 2401 = 7^4 \).
\( \implies \frac{3^4}{7^4} = \left(\frac{3}{7}\right)^4 \).
In simple words: Break down each number into prime factors first. Then group the same factors together and write them as powers.

Exam Tip: Always use prime factorization to break down composite numbers. This helps you find matching bases easily.

Question 5. Simplify.
a. \( (-3)^2 \times (-5)^2 \)
b. \( (-2)^4 \times (-7)^3 \)
c. \( (-1)^{10} \times (2)^5 \)
d. \( (-2)^3 \times (-3)^2 \)
Answer:
a. Solve each part: \( (-3)^2 = 9 \) and \( (-5)^2 = 25 \).
\( \implies 9 \times 25 = 225 \).
b. Solve each part: \( (-2)^4 = 16 \) and \( (-7)^3 = -343 \).
\( \implies 16 \times (-343) = -5488 \).
c. Solve each part: \( (-1)^{10} = 1 \) and \( (2)^5 = 32 \).
\( \implies 1 \times 32 = 32 \).
d. Solve each part: \( (-2)^3 = -8 \) and \( (-3)^2 = 9 \).
\( \implies -8 \times 9 = -72 \).
In simple words: Find the value of each power first. Then multiply those values to find the final number.

Exam Tip: A negative number raised to an even power becomes positive. A negative number raised to an odd power stays negative.

Question 6. Identify the greater number.
a. \( 2^6 \) or \( 6^2 \)
b. \( 2^{20} \) or \( 20^2 \)
c. \( 7^2 \) or \( 2^7 \)
d. \( 3^4 \) or \( 4^3 \)
Answer:
a. Calculate both values: \( 2^6 = 64 \) and \( 6^2 = 36 \).
Since \( 64 > 36 \), \( 2^6 \) is greater.
b. Calculate both values: \( 2^{20} \) is a very large number, while \( 20^2 = 400 \).
Since \( 2^{20} \) is much larger than 400, \( 2^{20} \) is greater.
c. Calculate both values: \( 7^2 = 49 \) and \( 2^7 = 128 \).
Since \( 128 > 49 \), \( 2^7 \) is greater.
d. Calculate both values: \( 3^4 = 81 \) and \( 4^3 = 64 \).
Since \( 81 > 64 \), \( 3^4 \) is greater.
In simple words: Work out the values of both numbers. Then compare them to see which one is larger.

Exam Tip: Never assume that a larger base automatically means a larger value. Always calculate both powers to be sure.

Question 7. Simplify and express in exponential form.
a. \( 2^3 \times 3^3 \)
b. \( 2^5 \div 2^2 \)
c. \( 9^{10} \div 9^7 \)
d. \( (2^3)^2 \times 2^4 \)
e. \( (5^{10} \div 5^7) \times 5^5 \)
f. \( (2^0 + 3^0) \times 5^0 \)
g. \( \frac{3 \times 7^2 \times 11^8}{21 \times 11^3} \)
h. \( \frac{108 \times 25^2}{2^6 \times 5^2} \)
i. \( \frac{4 \times 7^4 \times 3^5}{21 \times 2} \)
j. \( \frac{(a^3)^2 \times (b^2)^3}{a^2 \times b^3} \)
k. \( (2^0 \times 3^0 + 3^0 \times 4^0) \times 3^0 \)
l. \( \frac{3^5 \times 10^5 \times 25}{5^7 \times 6^5} \)
m. \( \frac{4^7 \times t^3 \times s^4}{4^7 \times t \times s} \)
n. \( \frac{25 \times 125 \times t^7}{10^5 \times t^2} \)
Answer:
a. Since the exponents are the same, multiply the bases.
\( \implies (2 \times 3)^3 = 6^3 \).
b. Since the bases are the same, subtract the exponents.
\( \implies 2^{5-2} = 2^3 \).
c. Subtract the exponents for division.
\( \implies 9^{10-7} = 9^3 \).
d. First multiply the exponents inside the parenthesis.
\( \implies 2^{3 \times 2} \times 2^4 = 2^6 \times 2^4 \)
Now add the exponents.
\( \implies 2^{6+4} = 2^{10} \).
e. Divide inside the parenthesis first by subtracting the exponents.
\( \implies 5^{10-7} \times 5^5 = 5^3 \times 5^5 \)
Now add the exponents.
\( \implies 5^{3+5} = 5^8 \).
f. Any non-zero number to the power of 0 is 1.
\( \implies (1 + 1) \times 1 = 2 \times 1 = 2 \).
g. Break 21 down into prime factors.
\( 21 = 3 \times 7 \).
\( \implies \frac{3 \times 7^2 \times 11^8}{3 \times 7 \times 11^3} \)
Cancel the common terms.
\( \implies 7^{2-1} \times 11^{8-3} = 7 \times 11^5 \).
h. Factor each term first.
\( 108 = 2^2 \times 3^3 \) and \( 25^2 = (5^2)^2 = 5^4 \).
\( \implies \frac{2^2 \times 3^3 \times 5^4}{2^6 \times 5^2} \)
Subtract exponents for matching bases.
\( \implies 2^{2-6} \times 3^3 \times 5^{4-2} = 2^{-4} \times 3^3 \times 5^2 \) (or \( \frac{3^3 \times 5^2}{2^4} \)).
i. Break down 4 and 21.
\( 4 = 2^2 \) and \( 21 = 3 \times 7 \).
\( \implies \frac{2^2 \times 7^4 \times 3^5}{3 \times 7 \times 2} \)
Subtract exponents of matching bases.
\( \implies 2^{2-1} \times 7^{4-1} \times 3^{5-1} = 2 \times 7^3 \times 3^4 \).
j. Multiply the inner and outer exponents first.
\( \implies \frac{a^{3 \times 2} \times b^{2 \times 3}}{a^2 \times b^3} = \frac{a^6 \times b^6}{a^2 \times b^3} \)
Subtract the exponents.
\( \implies a^{6-2} \times b^{6-3} = a^4 b^3 \).
k. Use the rule that any non-zero number to the power of 0 is 1.
\( \implies (1 \times 1 + 1 \times 1) \times 1 = (1 + 1) \times 1 = 2 \).
l. Write 10, 25, and 6 as prime factors.
\( 10^5 = (2 \times 5)^5 = 2^5 \times 5^5 \)
\( 25 = 5^2 \)
\( 6^5 = (2 \times 3)^5 = 2^5 \times 3^5 \)
\( \implies \frac{3^5 \times 2^5 \times 5^5 \times 5^2}{5^7 \times 2^5 \times 3^5} = \frac{2^5 \times 3^5 \times 5^7}{2^5 \times 3^5 \times 5^7} = 1 \).
m. Group similar bases together.
\( \implies 4^{7-7} \times t^{3-1} \times s^{4-1} = 4^0 \times t^2 \times s^3 = t^2 s^3 \).
n. Write 25, 125, and 10 as powers.
\( \implies \frac{5^2 \times 5^3 \times t^7}{(2 \times 5)^5 \times t^2} = \frac{5^5 \times t^7}{2^5 \times 5^5 \times t^2} \)
Cancel the common terms.
\( \implies \frac{t^{7-2}}{2^5} = \frac{t^5}{2^5} \) (or \( \left(\frac{t}{2}\right)^5 \)).
In simple words: Use exponent rules like adding powers for multiplying and subtracting powers for dividing. Make sure to combine only matching bases.

Exam Tip: Whenever you see a base raised to power 0, change it to 1 immediately to make the expression much simpler.

Question 8. Express the following in expanded form.
a. \( 7000932 \)
b. \( 5976080 \)
c. \( 1089234 \)
Answer:
a. We write the number using powers of 10 based on place values.
\( \implies 7000932 = 7 \times 10^6 + 0 \times 10^5 + 0 \times 10^4 + 0 \times 10^3 + 9 \times 10^2 + 3 \times 10^1 + 2 \times 10^0 \).
b. Write each digit multiplied by its place value as a power of 10.
\( \implies 5976080 = 5 \times 10^6 + 9 \times 10^5 + 7 \times 10^4 + 6 \times 10^3 + 0 \times 10^2 + 8 \times 10^1 + 0 \times 10^0 \).
c. Expand using powers of 10 based on each digit's place.
\( \implies 1089234 = 1 \times 10^6 + 0 \times 10^5 + 8 \times 10^4 + 9 \times 10^3 + 2 \times 10^2 + 3 \times 10^1 + 4 \times 10^0 \).
In simple words: Write out each digit multiplied by its place value. Use powers of 10 for the place values, starting from 0 on the far right.

Exam Tip: Do not skip the zero digits in your expansion. Always include them multiplied by their correct power of 10 to get full marks.

Question 9. Express the numbers in standard form.
a. \( 72586.239 \)
b. \( 923.45 \)
c. \( 496752000 \)
d. \( 1000000000000 \)
e. \( 1530000 \)
Answer:
a. Move the decimal 4 places to the left to leave one non-zero digit on the left.
\( \implies 7.2586239 \times 10^4 \).
b. Move the decimal 2 places to the left.
\( \implies 9.2345 \times 10^2 \).
c. Move the decimal point 8 places to the left.
\( \implies 4.96752 \times 10^8 \).
d. Move the decimal point 12 places to the left.
\( \implies 1 \times 10^{12} \).
e. Move the decimal point 6 places to the left.
\( \implies 1.53 \times 10^6 \).
In simple words: Put the decimal point after the first non-zero number. Then multiply by 10 raised to the power of how many places you moved the decimal.

Exam Tip: In standard form, the first number must always be between 1.0 and 9.999... (greater than or equal to 1, and strictly less than 10).