Class 7 Exponents Exercise 6.1

\begin{array}{l} \text {Q1. Find the value of each of the following: } \\ \text {(i) } \quad 13^{2} \quad \text {(ii) } \quad 7^{3} \quad \text {(iii) } \quad 3^{4} \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) } \quad 13^{2} = 13 \times 13 = 169 \\ \\\text {(ii) } \quad 7^{3} = 7 \times 7 \times 7 = 343 \\ \\ \text {(iii) } \quad 3^{4} = 3 \times 3 \times 3 \times 3 =81 \\ \end{array}\begin{array}{l} \text {Q2. Find the value of each of the following: } \\ \text {(i) } \quad (-7)^{2} \quad \text {(ii) } \quad (-3)^{4} \quad \text {(iii) } \quad (-5)^{5} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) We know that } (-x)^n = \text {Positive value, if n is even and } \\ \text { a negative value if n is odd.} \\ \\ \quad (-7)^{2} = (-7) \times (-7) = 49 \\ \\\text {(ii) } \quad (-3)^{4} = (-3) \times (-3) \times (-3) \times (-3) = 81 \\ \\\text {(iii) } \quad (-5)^{5} = (-5) \times (-5) \times (-5) \times (-5) \times (-5) = -3125 \\ \\\end{array}\begin{array}{l} \text {Q3. Simplify: } \\ \text {(i) } \quad 3 \times 10^{2} \quad \text {(ii) } \quad 2^{2} \times 5^{3} \quad \text {(iii) } \quad 3^{3} \times 5^{2} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad 3 \times 10^{2} = 3 \times 10 \times 10 = 300 \\ \\\text {(ii) } \quad 2^{2} \times 5^{3} =2 \times 2 \times 5 \times 5 \times 5 =500 \\ \\\text {(iii) } \quad 3^{3} \times 5^{2} = 3 \times 3 \times 3 \times 5 \times 5 \\ =675 \\\end{array}\begin{array}{l} \text {Q4. Simplify: } \\ \text {(i) } \quad 3^{2} \times 10^{4} \quad \text {(ii) } \quad 2^{4} \times 3^{2} \quad \text {(iii) } \quad 5^{2} \times 3^{4} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad 3^{2} \times 10^{4} =3 \times 3 \times 10 \times 10 \times 10 \times 10 = 90000 \\ \\\text {(ii) } \quad 2^{4} \times 3^{2} =2 \times 2 \times 2 \times 2 \times 3 \times 3 \\ =144 \text {(iii) } \quad 5^{2} \times 3^{4} =5 \times 5 \times 3 \times 3 \times 3 \times 3 \\ =2025 \\\end{array}\begin{array}{l} \text {Q5. Simplify: } \\ \text {(i) } \quad (-2) \times (-3)^{3} \quad \text {(ii) } \quad (-3)^{2} \times (-5)^{3} \quad \text {(iii) } \quad (-2)^{5} \times (-10)^{2} \end{array}\begin{array}{l} \text {Sol. We know that } (-x)^n = \text {Positive value, if n is even and } \\ \text { a negative value if n is odd.} \\ \\\text {(i) } \quad (-2) \times(-3)^{3} = (-2) \times(-3) \times(-3) \times(-3) =54 \\ \\\text {(ii) } \quad (-3)^{2} \times(-5)^{3} = (-3) \times(-3) \times(-5) \times(-5) \times(-5) \\ = -1125 \\ \\\text {(iii) } \quad (-2)^{5} \times(-10)^{2} =(-2) \times(-2) \times(-2) \times(-2) \times(-2) \times(-10) \times(-10) \\ =-3200 \end{array}\begin{array}{l} \text {Q6. Simplify: } \\ \text {(i) } \quad (\frac{3}{4})^{2} \quad \text {(ii) } \quad (\frac{-2}{3})^{4} \quad \text {(iii) } \quad (\frac{-4}{5})^{5} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad (\frac{3}{4})^{2} =\frac{3}{4} \times \frac{3}{4} \\ = \frac{9}{16} \\ \\\text {(ii) } \quad (\frac{-2}{3})^{4} =\frac{-2}{3} \times \frac{-2}{3} \times \frac{-2}{3} \times \frac{-2}{3} \\ =\frac{16}{81} \\ \\\text {(iii) } \quad (\frac{-4}{5})^{5}=\frac{-4}{5} \times \frac{-4}{5} \times \frac{-4}{5} \times \frac{-4}{5} \times \frac{-4}{5} \\ =\frac{-1024}{3125} \end{array}\begin{array}{l} \text {Q7. Identify the greater number in each of the following: } \\ \text {(i) } \quad 2^{5} or 5^{2} \quad \text {(ii) } \quad 3^{4} or 4^{3} \quad \text {(iii) } \quad 3^{5} or 5^{3} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad 2^{5}=2 \times 2 \times 2 \times 2 \times 2 = 32 \\ 5^{2}=5 \times 5 =25 \\ \\ \therefore \quad 2^{5} \gt 5^{2} \\ \\\text {(ii) } \quad 3^{4}=3 \times 3 \times 3 \times 3=81 \\ 4^{3}=4 \times 4 \times 4=64 \\ \\\therefore \quad 3^{4} \gt 4^{3} \\ \\\text {(iii) } \quad 3^{5}=3 \times 3 \times 3 \times 3 \times 3 =243 \\ 5^{3}=5 \times 5 \times 5 =125 \\ \\\therefore \quad 3^{5} \gt 5^{3}\end{array}\begin{array}{l} \text {Q8. Express each of the following in exponential form: } \\ \text {(i) } \quad (-5) \times(-5) \times(-5) \quad \text {(ii) } \quad \frac{-5}{7} \times \frac{-5}{7} \times \frac{-5}{7} \times \frac{-5}{7} \quad \text {(iii) } \quad \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad (-5) \times (-5) \times (-5) = (-5)^{3} \\ \\\text {(ii) } \quad \frac{-5}{7} \times \frac{-5}{7} \times \frac{-5}{7} \times \frac{-5}{7} = (\frac{-5}{7})^{4} \\ \\\text {(iii) } \quad \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} = (\frac{4}{3})^{5}\end{array}\begin{array}{l} \text {Q9. Express each of the following in exponential form: } \\ \text {(i) } \quad x \times x \times x \times x \times a \times a \times b \times b \times b \\ \text {(ii) } \quad (-2) \times(-2) \times(-2) \times(-2) \times a \times a \times a \\ \text {(iii) } \quad (\frac{-2}{3}) \times(\frac{-2}{3}) \times x \times x \times x \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) } \quad x \times x \times x \times x \times a \times a \times b \times b \times b = x^{4}a^{2}b^{3} \\ \\\text {(ii) } \quad (-2) \times(-2) \times(-2) \times(-2) \times a \times a \times a = (-2)^{4}a^{3} \\ \\ \text {(iii) } \quad (\frac{-2}{3}) \times(\frac{-2}{3}) \times x \times x \times x = = (\frac{-2}{3})^{2}x^{3} \\ \\\end{array}\begin{array}{l} \text {Q10. Express each of the following numbers in exponential form: } \\ \text {(i) } \quad 512 \quad \text {(ii) } \quad 625 \quad \text {(iii) } \quad 729 \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) Prime factorization of 512 } =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ =2^{9} \\ \\\text {(ii) Prime factorization of 625 } =5 \times 5 \times 5 \times 5 =5^{4} \\ \\\text {(iii) Prime factorization of 729 }=3 \times 3 \times 3 \times 3 \times 3 \times 3 \\ =3^{6} \\ \\\end{array}\begin{array}{l} \text {Q11. Express each of the following numbers as a product of powers of their prime factors: } \\ \text {(i) } \quad 36 \quad \text {(ii) } \quad 675 \quad \text {(iii) } \quad 392\end{array}\begin{array}{l} \text {Sol. } \\\text {(i) Prime factorization of 36 } =2 \times 2 \times 3 \times 3=2^{2} \times 3^{2} \\ \\\text {(ii) Prime factorization of 675 } =3 \times 3 \times 3 \times 5 \times 5 \\ =3^{3} \times 5^{2} \\ \\\text {(iii) Prime factorization of 392 } =2 \times 2 \times 2 \times 7 \times 7 \\ =2^{3} \times 7^{2} \\ \\\end{array}\begin{array}{l} \text {Q12. Express each of the following numbers as a product of powers of their prime factors: } \\ \text {(i) } \quad 450 \quad \text {(ii) } \quad 2800 \quad \text {(iii) } \quad 24000 \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) Prime factorization of 450 } =2 \times 3 \times 3 \times 5 \times 5 \\ =2 \times 3^{2} \times 5^{2} \\ \\\text {(ii) Prime factorization of 2800 } =2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 7 \\ =2^{4} \times 5^{2} \times 7 \\ \\\text {(iii) Prime factorization of 24000 } =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 \times 5 \\ =2^{6} \times 3 \times 5^{3} \\ \\\end{array}\begin{array}{l} \text {Q13. Express each of the following as a rational number of the form } \frac{p}{q} \\ \text {(i) } \quad (\frac{3}{7})^{2} \quad \text {(ii) } \quad (\frac{7}{9})^{3} \quad \text {(iii) } \quad (\frac{-2}{3})^{4} \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) } \quad (\frac{3}{7})^{2}=(\frac{3}{7}) \times (\frac{3}{7}) \\ =\frac{9}{49} \\ \\\text {(ii) } \quad (\frac{7}{9})^{3} =(\frac{7}{9}) \times (\frac{7}{9}) \times (\frac{7}{9}) \\ =\frac{343}{729} \\ \\\text {(iii) } \quad (\frac{-2}{3})^{4} =(\frac{-2}{3}) \times (\frac{-2}{3}) \times (\frac{-2}{3}) \times (\frac{-2}{3}) \\ =\frac{16}{81}\end{array}\begin{array}{l} \text {Q14. Express each of the following rational numbers in power notation: } \\ \text {(i) } \quad \frac{49}{64} \quad \text {(ii) } \quad -\frac{64}{125} \quad \text {(iii) } \quad -\frac{1}{216} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad \frac{49}{64} = \frac{7 \times 7}{8 \times 8} \\ = \frac{7}{8} \times \frac{7}{8} = (\frac{7}{8})^{2} \\ \\\text {(ii) } \quad -\frac{64}{125} = – \frac{4 \times 4 \times 4}{5 \times 5 \times 5} \\ =- \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \\ =- (\frac{4}{5})^{3} \\ = (\frac{-4}{5})^{3} \\ \\\text {(iii) } \quad -\frac{1}{216} = -\frac{1}{6 \times 6 \times 6} \\ = – (\frac{1}{6})^{3} \\ = (\frac{-1}{6})^{3}\end{array}\begin{array}{l} \text {Q15. Find the value of each of the following: } \\ \text {(i) } \quad (\frac{-1}{2})^{2} \times 2^{3} \times(\frac{3}{4})^{2} \\ \text {(ii) } \quad (\frac{-3}{5})^{4} \times(\frac{4}{9})^{4} \times(\frac{-15}{18})^{2} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad (\frac{-1}{2})^{2} \times 2^{3} \times(\frac{3}{4})^{2} \\ =\frac{1}{4} \times 8 \times \frac{9}{16} \\ = \frac{9}{8} \\ \\\text {(ii) } \quad (\frac{-3}{5})^{4} \times(\frac{4}{9})^{4} \times(\frac{-15}{18})^{2} \\ =\frac{81}{625} \times \frac{256}{6561} \times \frac{225}{324} \\ =\frac{64}{18225} \end{array}\begin{array}{l} \text {Q16. If a=2 and b=3, then find the values of each of the following:} \\ \text {(i) } \quad (a+b)^{a} \quad \text {(ii) } \quad (a b)^{b} \quad \text {(iii) } \quad (\frac{b}{a})^{b} \quad \text {(iv) } \quad (\frac{a}{b}+\frac{b}{a})^{a} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad (a+b)^{a} = (2+3)^{2} =5^{2}=25 \\ \\\text {(ii) } \quad (a b)^{b} =(2 \times 3)^{3} =6^{3}=216 \\ \\\text {(iii) } \quad (\frac{b}{a})^{b} = (\frac {3}{2})^{3} = \frac {27}{8} \\ \\\text {(iv) } \quad (\frac{a}{b}+\frac{b}{a})^{a} \\ =(\frac {2}{3}+\frac {3}{2})^{2} \\ =(\frac {2 \times 2 + 3 \times 3}{3 \times 2})^{2} \\ =(\frac {13}{6})^{2} \\ =\frac {169}{36} \\\end{array}