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Searching for NCERT Class 9 Maths Chapter 1 Exercise 1.6 solutions? You’re in the right place! This section offers clear, step-by-step solutions to all the questions in Exercise 1.6 of Chapter 1 – Number Systems. This exercise focuses on applying the concepts of irrational numbers, their properties, and simplifying expressions involving them. It helps students recognize how irrational numbers behave under various mathematical operations and how they fit into the real number system. These solutions aim to strengthen your understanding and boost your confidence in handling complex number problems. Perfect for exam preparation and concept reinforcement, this guide makes learning about irrational numbers easy, logical, and accessible for all learners.
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Exercise 1.6
1.
(i) Find: \( 64 \frac{1}{2} \)
Answer
\( (64)^{\frac{1 }{ 2}} \)Writing Prime factors of 64 we get that : \( 64=8 \times 8 \)
So, \((64)^{\frac{1 }{ 2}}=\left(2^{6}\right)^{\frac{1 }{ 2}}\)
\((64)^{\frac{1 }{ 2}}=2^{3}\)
\(=8\)
(ii) Find: \( 32^{\frac{1}{5}} \)
Answer
(ii) \( (32)^{\frac{1 }{ 5}} \)Writing Prime Factors of 32 we get that: \( 32=2 \times 2 \times 2 \times 2 \times 2 \)
So, \( 32=2^{5} \)
\( (32)^{\frac{1 }{ 5}}=\left(2^{5}\right)^{\frac{1 }{ 5}} \)
\( (32)^{\frac{1 }{ 5}}=2 \)
(iii) Find: \( 125^{\frac{1}{3}} \)
Answer
(iii) \( (125)^{\frac{1 }{ 3}} \) Writing Prime Factors of 125 we get that: \( 125=5 \times 5 \times 5 \)
So, \( 125=5^{3} \)
\( (125)^{\frac{1 }{ 3}}=\left(5^{3}\right)^{\frac{1 }{ 3}} \)
\( (125)^{\frac{1 }{ 3}}=5 \)
\( =5 \)
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2.
(i) Find: \( 9^{\frac{3}{2}} \)
Answer
\( 9^{\frac{3 }{ 2}}=\left(3^{2}\right)^{\frac{3 }{ 2}} \)\(=32 \times \frac{3 }{ 2}\)
\(=3^{3}\)
\(=27\)
(ii) Find: \( 32^{\frac{2}{5}} \)
Answer
\( 32^{\frac{2 }{ 5}}=\left(2^{5}\right)^{\frac{2 }{ 5}} \)\(=(2)^{5} \times \frac{2 }{ 5}\)
\(=2^{2}\)
\(=4\)
(iii) Find: \( 16^{\frac{3}{4}} \)
Answer
\( (16)^{\frac{3 }{ 4}}=\left(2^{4}\right)^{\frac{3 }{ 4}} \)\(=2^{3}\)
\( =8 \)
(iv) Find: \( 125^{\frac{-1}{3}} \)
Answer
\( (125)^{-\frac{1 }{ 3}}=\frac{1}{\sqrt[3]{125}} \)\( =\frac{1}{5} \)
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3.
(i) Simplify: \( 2^{\frac{2}{3}} 2^{\frac{1}{5}} \)
Answer
\( a^{m} \cdot a^{n}=a^{(m+n)} \)\( =2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}=2^{\frac{2}{3}+\frac{1}{5}} \)
\( =2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}=2^{\frac{(2 \times 5+1 \times 3)}{15}} \)
\( =2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}=2^{\frac{(2 \times 5+1 \times 3)}{15}} \)
\( =2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}=2^{\frac{13}{15}} \)
(ii) Simplify: \( \left(\frac{1}{3^{3}}\right)^{7} \)
Answer
\( \left(a^{m}\right)^{n}=a^{m n} \)
\(\left(\frac{1}{3^{3}}\right)^{7}=\frac{1}{3^{21}}\)
\(\left(\frac{1}{3^{3}}\right)=3^{-21}\)
(iii) Simplify: \( \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}
\)
Answer
\( \frac{a^{m}}{a^{n}}=a^{(m-n)}\)\(\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}=11^{\left(\frac{1}{2}-\frac{1}{4}\right)}\)
\(\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}=11^{\left(\frac{2-1}{4}\right)}\)
\(\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}=11^{\left(\frac{1}{4}\right)}\)
(iv) Simplify: \( 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} \)
Answer
\(\text { } a^{m} \cdot b^{m}=(a \cdot b)^{m}\)\(=7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}=(7 \times 8)^{\frac{1}{2}}\)
\(=7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}=56^{\frac{1}{2}}\)
