CBSE Class 7 Mathematics Exponents and Powers Assignment Set 05

VII - Mathematics Assignment No. 04 - Exponents and Powers

Simplify

Question 1. \((18)^{\frac{1}{3}} \times (768)^{\frac{1}{3}}\)
Answer: Prime factorize 18 and 768:
\(18 = 2 \times 3^2\)
\(768 = 3 \times 256 = 3 \times 2^8\)
Substitute into the expression:
\(= (2 \times 3^2)^{\frac{1}{3}} \times (3 \times 2^8)^{\frac{1}{3}}\)
Using the product rule of exponents \((ab)^n = a^n b^n\):
\(= 2^{\frac{1}{3}} \times 3^{\frac{2}{3}} \times 3^{\frac{1}{3}} \times 2^{\frac{8}{3}}\)
Combine terms with the same base by adding their exponents:
For base 2: \(2^{\frac{1}{3} + \frac{8}{3}} = 2^{\frac{9}{3}} = 2^3 = 8\)
For base 3: \(3^{\frac{2}{3} + \frac{1}{3}} = 3^{\frac{3}{3}} = 3^1 = 3\)
Multiply the results:
\(= 8 \times 3 = 24\)
In simple words: Break down the numbers into prime factors and combine powers with the same base.

Exam Tip: Always prime factorize composite numbers under fractional exponents to easily apply exponent laws.

Question 2. \((125)^{-\frac{2}{3}} \times (27)^{-\frac{2}{3}}\)
Answer: Express 125 and 27 as powers of primes:
\(125 = 5^3\)
\(27 = 3^3\)
Substitute into the expression:
\(= (5^3)^{-\frac{2}{3}} \times (3^3)^{-\frac{2}{3}}\)
Use the power of a power rule \((a^m)^n = a^{m \times n}\):
\(= 5^{3 \times (-\frac{2}{3})} \times 3^{3 \times (-\frac{2}{3})}\)
\(= 5^{-2} \times 3^{-2}\)
Rewrite using positive exponents \(a^{-n} = \frac{1}{a^n}\):
\(= \frac{1}{5^2} \times \frac{1}{3^2}\)
\(= \frac{1}{25} \times \frac{1}{9} = \frac{1}{225}\)
In simple words: Convert the bases to exponents, multiply the powers, and flip negative exponents to the denominator.

Exam Tip: Remember that \((a^m)^n = a^{m \cdot n}\). The denominator 3 in the exponent easily cancels out the cube power of 125 and 27.

Question 3. \((0.000729)^{\frac{5}{6}}\)
Answer: Convert the decimal to a fraction:
\(0.000729 = \frac{729}{1000000} = \frac{3^6}{10^6} = \left(\frac{3}{10}\right)^6\)
Substitute back into the expression:
\(= \left[\left(\frac{3}{10}\right)^6\right]^{\frac{5}{6}}\)
Multiply the exponents:
\(= \left(\frac{3}{10}\right)^{6 \times \frac{5}{6}}\)
\(= \left(\frac{3}{10}\right)^5\)
Calculate the value:
\(= \frac{243}{100000} = 0.00243\)
In simple words: Change the decimal to a fraction with a power of 6, then multiply the outer exponent.

Exam Tip: Count the decimal places carefully (6 places) to determine that the denominator is \(10^6\).

Question 4. \(\left[(51)^{\frac{1}{2}}\right]^4\)
Answer: Use the power of a power rule \((a^m)^n = a^{m \times n}\):
\(= 51^{\frac{1}{2} \times 4}\)
\(= 51^2\)
Calculate the square of 51:
\(51 \times 51 = 2601\)
In simple words: Multiply the fractional power by 4, then evaluate the resulting square.

Exam Tip: Directly multiply the exponents when an exponent is raised to another power.

Question 5. \((64)^{\frac{1}{3}} \times (64)^{\frac{2}{3}} \times (64)^{-\frac{4}{3}}\)
Answer: Since all bases are the same, use the product rule by adding the exponents: \(a^m \times a^n \times a^p = a^{m + n + p}\)
\(= 64^{\frac{1}{3} + \frac{2}{3} + \left(-\frac{4}{3}\right)}\)
Add the fractions in the exponent:
\(= 64^{\frac{1 + 2 - 4}{3}}\)
\(= 64^{-\frac{1}{3}}\)
Express 64 as \(4^3\):
\(= (4^3)^{-\frac{1}{3}}\)
Multiply the exponents:
\(= 4^{3 \times \left(-\frac{1}{3}\right)}\)
\(= 4^{-1} = \frac{1}{4}\)
In simple words: Combine all exponents with the same base by adding them, then simplify.

Question 6. \((125)^{-\frac{2}{3}} \times (64)^{\frac{4}{3}}\)
Answer: Express 125 and 64 as powers of primes:
\(125 = 5^3\)
\(64 = 4^3\)
Substitute into the expression:
\(= (5^3)^{-\frac{2}{3}} \times (4^3)^{\frac{4}{3}}\)
Multiply the exponents:
\(= 5^{3 \times \left(-\frac{2}{3}\right)} \times 4^{3 \times \frac{4}{3}}\)
\(= 5^{-2} \times 4^4\)
Rewrite with positive exponents and evaluate:
\(= \frac{1}{5^2} \times 256\)
\(= \frac{1}{25} \times 256 = \frac{256}{25}\)
In simple words: Write each base as a cube, apply the power rule to remove fractions, and compute.

Exam Tip: Keep fractions improper in final answers unless mixed fractions are specifically requested.

Question 7. \((32)^{-\frac{2}{5}} \times (125)^{\frac{2}{3}}\)
Answer: Express 32 and 125 in exponential form:
\(32 = 2^5\)
\(125 = 5^3\)
Substitute into the expression:
\(= (2^5)^{-\frac{2}{5}} \times (5^3)^{\frac{2}{3}}\)
Multiply the exponents:
\(= 2^{5 \times \left(-\frac{2}{5}\right)} \times 5^{3 \times \frac{2}{3}}\)
\(= 2^{-2} \times 5^2\)
Evaluate the powers:
\(= \frac{1}{2^2} \times 25\)
\(= \frac{1}{4} \times 25 = \frac{25}{4}\)
In simple words: Convert bases 32 and 125 to powers of 2 and 5 respectively, simplify exponents, and evaluate.

Question 8. \(7 \times 7^{\frac{3}{2}} \times 7^{-\frac{1}{2}}\)
Answer: Rewrite the first term 7 as \(7^1\):
\(= 7^1 \times 7^{\frac{3}{2}} \times 7^{-\frac{1}{2}}\)
Since bases are identical, add all exponents:
\(= 7^{1 + \frac{3}{2} - \frac{1}{2}}\)
Simplify the exponent fraction:
\(1 + \frac{3}{2} - \frac{1}{2} = 1 + \frac{2}{2} = 1 + 1 = 2\)
Substitute back:
\(= 7^2 = 49\)
In simple words: Combine terms with base 7 by adding their exponents together.

Exam Tip: A standalone number like 7 always has an invisible exponent of 1.

Question 9. \((729)^{\frac{1}{3}} \times (256)^{\frac{1}{4}}\)
Answer: Express 729 and 256 in exponential form:
\(729 = 9^3\)
\(256 = 4^4\)
Substitute into the expression:
\(= (9^3)^{\frac{1}{3}} \times (4^4)^{\frac{1}{4}}\)
Multiply the exponents:
\(= 9^{3 \times \frac{1}{3}} \times 4^{4 \times \frac{1}{4}}\)
\(= 9^1 \times 4^1\)
Multiply the values:
\(= 9 \times 4 = 36\)
In simple words: Find numbers that raised to the 3rd and 4th powers give 729 and 256, then simplify.

Exam Tip: Memorizing squares up to 30 and cubes up to 15 saves significant time on exponent problems.

Question 10. \((1^3 + 2^3 + 3^3)^{\frac{1}{2}}\)
Answer: Evaluate the cubes inside the parentheses first:
\(1^3 = 1\)
\(2^3 = 8\)
\(3^3 = 27\)
Sum the values inside the parentheses:
\(1 + 8 + 27 = 36\)
Apply the outer exponent \(\frac{1}{2}\) (which represents the square root):
\(= (36)^{\frac{1}{2}} = \sqrt{36} = 6\)
In simple words: Solve inside the bracket first following the order of operations, then apply the fractional power.

Exam Tip: Always evaluate operations inside grouping symbols (parentheses) before applying exponents outside.