MP Board 9th Laws of Exponents for Real Numbers वास्तविक संख्याओं के लिए घातांक-नियम

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1.5 वास्तविक संख्याओं के लिए घातांक-नियम (Laws of Exponents for Real Numbers)

MP Board 9th Laws of Exponents for Real Numbers : MP Board 9th Laws of Exponents for Real Numbers वास्तविक संख्याओं के लिए घातांक-नियम

प्रश्नावली 1.5 (Exercise 1.5)

1. ज्ञात कीजिए (Find the value of):

(i) 64^{\frac{1}{2}}

  • Solution: We know that 64 = 8^2.

        \[(8^2)^{\frac{1}{2}} = 8^{2 \times \frac{1}{2}} = 8^1 = 8\]

(ii) 32^{\frac{1}{5}}

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  • Solution: We know that 32 = 2^5.

        \[(2^5)^{\frac{1}{5}} = 2^{5 \times \frac{1}{5}} = 2^1 = 2\]

(iii) 125^{\frac{1}{3}}

  • Solution: We know that 125 = 5^3.

        \[(5^3)^{\frac{1}{3}} = 5^{3 \times \frac{1}{3}} = 5^1 = 5\]

2. ज्ञात कीजिए (Find the value of):

(i) 9^{\frac{3}{2}}

  • Solution: We know that 9 = 3^2.

        \[(3^2)^{\frac{3}{2}} = 3^{2 \times \frac{3}{2}} = 3^3 = 27\]

(ii) 32^{\frac{2}{5}}

  • Solution: We know that 32 = 2^5.

        \[(2^5)^{\frac{2}{5}} = 2^{5 \times \frac{2}{5}} = 2^2 = 4\]

(iii) 16^{\frac{3}{4}}

  • Solution: We know that 16 = 2^4.

        \[(2^4)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}} = 2^3 = 8\]

(iv) 125^{\frac{-1}{3}}

  • Solution: We know that 125 = 5^3. Using the rule a^{-m} = \frac{1}{a^m}.

        \[(5^3)^{\frac{-1}{3}} = 5^{3 \times \frac{-1}{3}} = 5^{-1} = \frac{1}{5}\]

3. सरल कीजिए (Simplify):

(i) 2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}

  • Solution: Using the rule a^m \cdot a^n = a^{m+n}.

        \[2^{\frac{2}{3} + \frac{1}{5}} = 2^{\frac{10+3}{15}} = 2^{\frac{13}{15}}\]

(ii) (\frac{1}{3^3})^7

  • Solution: Using the rule (a^m)^n = a^{m \times n}.

        \[\frac{1^7}{(3^3)^7} = \frac{1}{3^{3 \times 7}} = \frac{1}{3^{21}} \quad \text{or} \quad 3^{-21}\]

(iii) \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}

  • Solution: Using the rule \frac{a^m}{a^n} = a^{m-n}.

        \[11^{\frac{1}{2} - \frac{1}{4}} = 11^{\frac{2-1}{4}} = 11^{\frac{1}{4}}\]

(iv) 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}

  • Solution: Using the rule a^m \cdot b^m = (ab)^m.

        \[(7 \times 8)^{\frac{1}{2}} = 56^{\frac{1}{2}}\]