Odisha State Board BSE Odisha 8th Class Maths Solutions Algebra Chapter 6 ବର୍ଗ-ବର୍ଗମୂଳ ଏବଂ ଘନ-ଘନମୂଳ Ex 6(h) Textbook Exercise Questions and Answers.
BSE Odisha Class 8 Maths Solutions Algebra Chapter 6 ବର୍ଗ-ବର୍ଗମୂଳ ଏବଂ ଘନ-ଘନମୂଳ Ex 6(h)
Question 1.
ଘନ ନିର୍ଣ୍ଣୟ କର ।
(i) \(\frac{7}{9}\)
(ii) \(– \frac{8}{11}\)
(iii) \(\frac{12}{7}\)
(iv) \(– \frac{13}{8}\)
(v) \(2 \frac{3}{5}\)
(vi) \(3 \frac{1}{4}\)
(vii) \(-1 \frac{2}{3}\)
(viii) 0.2
(ix) 1.3
(x) 0.03
ସମାଧାନ :
(i) \(\frac{7}{9}\)ର ଘନ = \(\left(\frac{7}{9}\right)^3=\frac{7^3}{9^3}=\frac{343}{729}\)
(ii) \(– \frac{8}{11}\)ର ଘନ = \(\left(\frac{-8}{11}\right)^3=\frac{(-8)^3}{11^3}=-\frac{512}{1331}\)
(iii) \(\frac{12}{7}\)ର ଘନ = \(\left(\frac{12}{7}\right)^3=\frac{12^3}{7^3}=\frac{1728}{343}\)
(iv) \(– \frac{13}{8}\)ର ଘନ = \(\left(\frac{-13}{8}\right)^3=\frac{(-13)^3}{8^3}=\frac{-2197}{512}\)
(v) \(2 \frac{3}{5}\)ର ଘନ = \(\left(2 \frac{3}{5}\right)^3=\left(\frac{13}{5}\right)^3=\frac{13^3}{5^3}=\frac{2197}{125}\)
(vi) \(3 \frac{1}{4}\)ର ଘନ = \(\left(3 \frac{1}{4}\right)^3=\left(\frac{13}{4}\right)^3=\frac{13^3}{4^3}=\frac{2197}{64}\)
(vii) \(-1 \frac{2}{3}\)ର ଘନ = \(\left(-1 \frac{2}{3}\right)^3=\left(-\frac{5}{3}\right)^3=\frac{(-5)^3}{3^3}=\frac{-125}{27}\)
(viii) 0.2ର ଘନ = \((0 \cdot 2)^3=\left(\frac{2}{10}\right)^3=\frac{2^3}{10^3}=\frac{8}{1000}=0.008\)
(ix) 1.3ର ଘନ = \((1 \cdot 3)^3=\left(\frac{13}{10}\right)^3=\frac{13^3}{10^3}=\frac{2197}{1000}=2 \cdot 197\)
(x) 0.03ର ଘନ = \((0 \cdot 03)^3=\left(\frac{3}{100}\right)^3=\frac{(3)^3}{(100)^3}=\frac{27}{1000000}=0 \cdot 000027\)
Question 2.
ଘନମୂଳ ନିର୍ଣ୍ଣୟ କର ।
(i) \(\frac{8}{125}\)
(ii) \(\frac{-64}{1331}\)
(iii) \(\frac{-27}{4096}\)
(iv) \(– \frac{2197}{9261}\)
(v) 0.001
(vi) 0.008
(vii) 1.728
(viii) 0.000125
ସମାଧାନ :
(i) \(\sqrt[3]{\frac{8}{125}}=\sqrt[3]{\frac{2^3}{5^3}}=\sqrt[3]{\left(\frac{2}{5}\right)^3}=\frac{2}{5}\)
(ii) \(\sqrt[3]{-\frac{64}{1331}}=\sqrt[3]{\frac{(-4)^3}{(11)^3}}=\sqrt[3]{\left(\frac{-4}{11}\right)^3}=\frac{-4}{11}\)
(iii) \(\sqrt[3]{\frac{-27}{4096}}=\sqrt[3]{\frac{(-3)^3}{(16)^3}}=\sqrt[3]{\left(\frac{-3}{16}\right)^3}=\frac{-3}{16}\)
(iv) \(\sqrt[3]{\frac{2197}{9261}}=\sqrt[3]{\frac{(13)^3}{(21)^3}}=\sqrt[3]{\left(\frac{13}{21}\right)^3}=\frac{13}{21}\)
(v) \(\sqrt[3]{0.001}=\sqrt[3]{\frac{1}{1000}}=\sqrt[3]{\frac{1^3}{10^3}}=\sqrt[3]{\left(\frac{1}{10}\right)^3}=\frac{1}{10}=0 \cdot 1\)
(vi) \(\sqrt[3]{0 \cdot 008}=\sqrt[3]{\frac{8}{1000}}=\sqrt[3]{\frac{2^3}{10^3}}=\sqrt[3]{\left(\frac{2}{10}\right)^3}=\frac{2}{10}=0 \cdot 2\)
(vii) \(\sqrt[3]{1 \cdot 728}=\sqrt[3]{\frac{1728}{1000}}=\sqrt[3]{\frac{12^3}{10^3}}=\sqrt[3]{\left(\frac{12}{10}\right)^3}=\frac{12}{10}=1 \cdot 2\)
(viii) \(\sqrt[3]{0 \cdot 000125}=\sqrt[3]{\frac{125}{1000000}}=\sqrt[3]{\frac{5^3}{(100)^3}}=\sqrt[3]{\left(\frac{5}{100}\right)^3}=\frac{5}{100}=0 \cdot 05\)
Question 3.
ନିମ୍ନୋକ୍ତ କେଉଁ ରାଶି କୌଣସି ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ଅଟେ ?
(i) \(\frac{27}{64}\)
(ii) \(\frac{125}{128}\)
(iii) \(\frac{-216}{729}\)
(iv) \(– \frac{-250}{686}\)
(v) 0.08
(vi) 0.125
(vii) 0.1331
ସମାଧାନ :
(m, n ∈ Q ଏବଂ n = m³ ହେଲେ, n ଏକ ଘନ ସଂଖ୍ୟା ହେବ ।)
(i) \(\frac{27}{64}=\frac{3 \times 3 \times 3}{4 \times 4 \times 4}=\left(\frac{3}{4}\right)^3\) ଏହା ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ।
(ii) \(\frac{125}{128}=\frac{5 \times 5 \times 5}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}=\frac{5^3}{2^3 \times 2^3 \times 2}\) ଏହା ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ନୁହେଁ ।
(iii) \(\frac{-216}{729}=\frac{(-6) \times(-6) \times(-6)}{9 \times 9 \times 9}=\left(\frac{-6}{9}\right)^3\) ଏହା ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ଅଟେ ।
(iv) \(– \frac{-250}{686}=\frac{-5 \times 5 \times 5 \times 2}{7 \times 7 \times 7 \times 2}=\left(\frac{-5}{7}\right)^3\) ଏହା ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ନୁହେଁ ।
(v) 0.08 = \(\frac{8}{10}=\frac{2 \times 2 \times 2}{2 \times 5}=\frac{(2)^3}{2 \times 5}\) ଏହା ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ଅଟେ ।
(vi) 0.125 = \(\frac{125}{1000}=\frac{5 \times 5 \times 5}{5 \times 5 \times 5 \times 2 \times 2 \times 2}=\frac{1}{2^3}=\left(\frac{1}{2}\right)^3\) ଏହା ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ଅଟେ ।
(vii) 0.1331 = \(\frac{1331}{10000}=\frac{11 \times 11 \times 11}{10 \times 10 \times 10 \times 10}=\frac{11^3}{10^3 \times 10}\) ଏହା ଏକ ପରିମେୟ ସଂଖ୍ୟାର ଘନ ନୁହେଁ ।