https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/v/some-special-cases-of-a-b-n 2025-03-01T15:52:37.13987578Z Some special cases of (a+b)^n We have learnt how to expand (a+b)^n in our previous videos. In this video, we'll replace a and b with different values to get some special cases and results. We first replace b with -b to get the expansion of (a-b)^n. We then put a = 1 and b = x in both to get two more special expansions - (1+x)^n and (1-x)^n. Finally, we plug x = 1 in both to get two special results regarding the coefficients. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Some special cases of (a+b)^n We have learnt how to expand (a+b)^n in our previous videos. In this video, we'll replace a and b with different values to get some special cases and results. We first replace b with -b to get the expansion of (a-b)^n. We then put a = 1 and b = x in both to get two more special expansions - (1+x)^n and (1-x)^n. Finally, we plug x = 1 in both to get two special results regarding the coefficients. https://cdn.kastatic.org/ka-youtube-converted/5g_koVnNvIQ.mp4/5g_koVnNvIQ.mp4 548 Applications of Binomial Theorem https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/e/some-special-cases-of-a-b-n 2025-05-31T02:32:31.283700489Z Some special cases of (a+b)^n Some special cases of (a+b)^n Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/v/worked-example-special-cases 2025-03-01T15:52:37.13987578Z Worked example (special cases) In this video, we'll apply what we know about expansions of (a+b)^n and (a-b)^n to solve problems. We'll first expand (x + 1/x)^6 and then (2x-3)^6 by carefully replacing as and bs with appropriate values. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Worked example (special cases) In this video, we'll apply what we know about expansions of (a+b)^n and (a-b)^n to solve problems. We'll first expand (x + 1/x)^6 and then (2x-3)^6 by carefully replacing as and bs with appropriate values. https://cdn.kastatic.org/ka-youtube-converted/016BgrrQ73A.mp4/016BgrrQ73A.mp4 441 Applications of Binomial Theorem https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/v/compute-powers-of-integers 2025-03-01T15:52:37.13987578Z Compute powers of integers We've learnt how to expand (a+b)^n and its special cases. In this video, we'll apply what we've learnt to find the values of powers of large numbers. We'll break them down into sums and then apply binomial expansion to get the results. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Compute powers of integers We've learnt how to expand (a+b)^n and its special cases. In this video, we'll apply what we've learnt to find the values of powers of large numbers. We'll break them down into sums and then apply binomial expansion to get the results. https://cdn.kastatic.org/ka-youtube-converted/6pj3Cxx5V7o.mp4/6pj3Cxx5V7o.mp4 563 Applications of Binomial Theorem https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/e/powers-of-integers-using-binomial-theorem 2025-05-31T02:32:31.283700489Z Powers of integers using Binomial Theorem Powers of integers using Binomial Theorem Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/v/using-binomial-theorem-to-compare-numbers 2025-03-01T15:52:37.13987578Z Using Binomial theorem to compare numbers We've learn't how to expand (a+b)^n and (1+x)^n. Let's use these expansions to compare powers of 1.1 and 1.01 with large numbers. We first expand 1.01^10,00,000 using (1+x)^n and compare the result with 10,000. We then expand 1.1^10,000 using (1+x)^n and compare the result with 5,00,000. In this video, we'll realise that we don't need lakhs of terms to figure things out, usually simplifying the first few terms does the job for us. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Using Binomial theorem to compare numbers We've learn't how to expand (a+b)^n and (1+x)^n. Let's use these expansions to compare powers of 1.1 and 1.01 with large numbers. We first expand 1.01^10,00,000 using (1+x)^n and compare the result with 10,000. We then expand 1.1^10,000 using (1+x)^n and compare the result with 5,00,000. In this video, we'll realise that we don't need lakhs of terms to figure things out, usually simplifying the first few terms does the job for us. https://cdn.kastatic.org/ka-youtube-converted/G2270V8gOwA.mp4/G2270V8gOwA.mp4 408 Applications of Binomial Theorem https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/e/comparing-numbers-using-binomial-theorem 2025-05-31T02:32:31.283700489Z Comparing numbers using Binomial Theorem Comparing numbers using Binomial Theorem Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/v/using-binomial-theorem-to-prove-statements 2025-03-01T15:52:37.13987578Z Using Binomial theorem to prove statements In this video, we apply binomial theorem to prove statements on divisibility. In such problems, we're given an expression and a statement that is true for all values of n. Our task is to strategically expand a binomial such that the expansion gives terms that match the given statement. We first prove that a given expression is always divisible by 64. We then prove that a given expression always leaves remainder 1 when divided by 25. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Using Binomial theorem to prove statements In this video, we apply binomial theorem to prove statements on divisibility. In such problems, we're given an expression and a statement that is true for all values of n. Our task is to strategically expand a binomial such that the expansion gives terms that match the given statement. We first prove that a given expression is always divisible by 64. We then prove that a given expression always leaves remainder 1 when divided by 25. https://cdn.kastatic.org/ka-youtube-converted/aH7oIpFPUuQ.mp4/aH7oIpFPUuQ.mp4 471 Applications of Binomial Theorem https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/e/binomial-theorem-applications 2025-05-31T02:32:31.283700489Z Binomial Theorem applications Binomial Theorem applications Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/v/using-binomial-theorem-irrational-terms 2025-03-01T15:52:37.13987578Z Using Binomial theorem (irrational terms) In this video, we'll apply binomial expansion on irrational terms. We'll consider pairs of expansions where almost half the terms cancel out because of symmetry. We'll use this technique to simplify the expressions. Finally, we'll replace the values of a, b, x, or y with appropriate irrational numbers to deduce the values of expressions given in the problem. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Using Binomial theorem (irrational terms) In this video, we'll apply binomial expansion on irrational terms. We'll consider pairs of expansions where almost half the terms cancel out because of symmetry. We'll use this technique to simplify the expressions. Finally, we'll replace the values of a, b, x, or y with appropriate irrational numbers to deduce the values of expressions given in the problem. https://cdn.kastatic.org/ka-youtube-converted/CuZBziJJazE.mp4/CuZBziJJazE.mp4 386 Applications of Binomial Theorem https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/e/applying-binomial-theorem-on-irrational-terms 2025-05-31T02:32:31.283700489Z Applying Binomial Theorem on irrational terms Applying Binomial Theorem on irrational terms Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/v/approximations-using-binomial-theorem 2025-05-22T11:09:14.016569349Z Approximations using binomial theorem In this video, we use binomial expansion to approximate powers of numbers close to 1. We use (1+x)^n for numbers greater than 1 and (1-x)^n for numbers less than 1. We realise through expansion that the first few terms matter much more than the remaining terms. So to approximate the value of these powers, we can ignore a lot of terms. We also realise that the approximations work better for smaller values of x. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Approximations using binomial theorem In this video, we use binomial expansion to approximate powers of numbers close to 1. We use (1+x)^n for numbers greater than 1 and (1-x)^n for numbers less than 1. We realise through expansion that the first few terms matter much more than the remaining terms. So to approximate the value of these powers, we can ignore a lot of terms. We also realise that the approximations work better for smaller values of x. https://cdn.kastatic.org/ka-youtube-converted/DQh0ex8m5hA.mp4/DQh0ex8m5hA.mp4 573 Applications of Binomial Theorem https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:binomial-theorem-ncert-new/x0419e5b3b578592a:applications-of-binomial-theorem/e/approximations-using-binomial-theorem 2025-05-31T02:32:31.283700489Z Approximations using Binomial Theorem Approximations using Binomial Theorem Ashish Gupta exercise