Rút gọn : \(\frac{a+b}{\sqrt{a}+\sqrt{b}}.\)
Rút gọn \(\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\frac{a\sqrt{a}}{\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a}}{\sqrt{a}+1}\)
\(\frac{a-b}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{a}^3-\sqrt{b}^3}{a+b+\sqrt{ab}}\).Rút gọn
\(\frac{a-b}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{a}^3-\sqrt{b}^3}{a+b+\sqrt{ab}}\)
\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}-\sqrt{b}}-\frac{\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)}{a+b+\sqrt{ab}}\)
\(=\sqrt{a}+\sqrt{b}-\left(\sqrt{a}-\sqrt{b}\right)\)
\(=\sqrt{a}-\sqrt{a}+\sqrt{b}+\sqrt{b}\)
\(=2\sqrt{b}\)
rút gọn:
\(\frac{a-b}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{a^3}-\sqrt{b^3}}{a-b}\)
Rút gọn biểu thức:
\(\left(\sqrt{a}+\frac{b-\sqrt{ab}}{\sqrt{a}+\sqrt{b}}\right)\div\left(\frac{a}{\sqrt{ab}+b}+\frac{b}{\sqrt{ab}-a}-\frac{a+b}{\sqrt{ab}}\right)\)
\(\left(\sqrt{a}+\frac{b-\sqrt{ab}}{\sqrt{a}+\sqrt{b}}\right)\div\left(\frac{a}{\sqrt{ab}+b}+\frac{b}{\sqrt{ab}-a}-\frac{a+b}{\sqrt{ab}}\right)\)
\(=\left(\frac{\sqrt{a}.\left(\sqrt{a}+\sqrt{b}\right)+b-\sqrt{ab}}{\sqrt{a}+\sqrt{b}}\right):\left(\frac{a}{\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)}+\frac{b}{\sqrt{a}\left(\sqrt{b}-\sqrt{a}\right)}-\frac{a+b}{\sqrt{ab}}\right)\)
\(=\left(\frac{a+\sqrt{ab}+b-\sqrt{ab}}{\sqrt{a}+\sqrt{b}}\right):\left(\frac{a.\sqrt{a}.\left(\sqrt{b}-\sqrt{a}\right)+b.\sqrt{b}.\left(\sqrt{a}+\sqrt{b}\right)-\left(a+b\right).\left(b-a\right)}{\sqrt{ab}.\left(b-a\right)}\right)\)
\(=\left(\frac{a+b}{\sqrt{a}+\sqrt{b}}\right):\left(\frac{a\sqrt{ab}-a^2+b\sqrt{ab}+b^2-b^2+a^2}{\sqrt{ab}.\left(b-a\right)}\right)\)
giải tiếp
\(=\left(\frac{a+b}{\sqrt{a}+\sqrt{b}}\right):\left(\frac{a\sqrt{ab}+b\sqrt{ab}}{\sqrt{ab}\left(b-a\right)}\right)\)
\(=\left(\frac{a+b}{\sqrt{a}+\sqrt{b}}\right):\left(\frac{\sqrt{ab}.\left(a+b\right)}{\sqrt{ab}.\left(b-a\right)}\right)=\left(\frac{a+b}{\sqrt{a}+\sqrt{b}}\right).\left(\frac{b-a}{a+b}\right)\)
\(=\frac{b-a}{\sqrt{a}+\sqrt{b}}=\frac{\left(b-a\right)\left(\sqrt{a}-\sqrt{b}\right)}{a-b}=\frac{b\sqrt{a}-b\sqrt{b}-a\sqrt{a}+a\sqrt{b}}{a-b}\)
Mình rút gọn tiếp theo kết quả bạn MMS Hồ Khánh Châu:
\(\frac{b\sqrt{a}-b\sqrt{b}-a\sqrt{a}+a\sqrt{b}}{a-b}.\)
\(=\frac{b\left(\sqrt{a}-\sqrt{b}\right)-a\left(\sqrt{a}-\sqrt{b}\right)}{a-b}\)
\(=\frac{\left(b-a\right)\left(\sqrt{a}-\sqrt{b}\right)}{a-b}\)
\(=\sqrt{b}-\sqrt{a}\)
A=\(\left(\frac{b-x}{\sqrt{b}-\sqrt{x}}-\frac{b\sqrt{b}-x\sqrt{x}}{b-x}\right).\frac{\left(\sqrt{b}+\sqrt{x}\right)^2}{b\sqrt{b}+x\sqrt{x}}\)
a/ Rút gọn
b/ So sánh A và \(\sqrt{A}\)
Em làm câu rút gọn =\(\frac{\sqrt{bx}}{b-\sqrt{bx}+x}\)
Nhưng câu b không biết so sánh sao? Các anh chị giúp em với !
Rút gọn biểu thức:
A= \(\left(\frac{1}{\sqrt{a}+\sqrt{b}}+\frac{3\sqrt{ab}}{a\sqrt{a}+b\sqrt{b}}\right).\left[\left(\frac{1}{\sqrt{a}-\sqrt{b}}-\frac{3\sqrt{ab}}{a\sqrt{a}-b\sqrt{b}}\right):\frac{a-b}{a+\sqrt{ab}+b}\right]\)
Rút gọn
\( A=\left(\frac{1}{\sqrt{a}-\sqrt{a-b}}+\frac{1}{\sqrt{a}+\sqrt{a+b}}\right)\div\left(1+\frac{\sqrt{a+b}}{\sqrt{a-b}}\right)\)
Ta có: a√a = √(a².a) = (√a)³
=> 1 - a√a = 1 - (√a)³ = (1 - √a)(a + √a + 1) (1)
Tương tự: 1 + a√a = 1 + (√a)³ = (1 + √a)(a - √a + 1) (2)
Từ (1) và (2) => [ (1-a√a/1-√a+√a).(1+a√a/1+√a-√a) + 1 ].
= [(1 - √a)(a + √a + 1)/(1 - √a) + √a].[(1 + √a)(a - √a + 1)/(1 + √a) - √a ] +1
=(a + √a + 1 + √a)(a - √a + 1- √a) + 1
= (a + 2√a + 1)(a - 2√a + 1) + 1
= (√a + 1)²(√a - 1)² +1
= [(√a + 1)(√a - 1)]² + 1
= (a - 1)² + 1
= a² - 2a + 1 + 1
= a² - 2a + 2
=> [ (1-a√a/1-√a+√a).(1+a√a/1+√a-√a) + 1 ] = a² - 2a + 2 (3)
Áp dụng (3) vào A ta được A = [(1 - a)²]/(a² - 2a + 2)
<=> A = (a² - 2a + 1)/(a² - 2a + 2)
rút gọn biểu thức
\(A=\frac{\left(\frac{1}{\sqrt{a}-\sqrt{a-b}}+\frac{1}{\sqrt{a}+\sqrt{a+b}}\right)}{\left(1+\frac{\sqrt{a+b}}{\sqrt{a-b}}\right)}\)
Rút gọn biểu thức \(\left[\frac{2\sqrt{ab}}{a-b}+\frac{\sqrt{a}-\sqrt{b}}{2\left(\sqrt{a}+\sqrt{b}\right)}\right].\frac{2\sqrt{a}}{\sqrt{a}+\sqrt{b}}+\frac{\sqrt{b}}{\sqrt{b}-\sqrt{a}}\)
\(=\left(\frac{2\sqrt{ab}}{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}+\frac{\sqrt{a}-\sqrt{b}}{2\left(\sqrt{a}+\sqrt{b}\right)}\right).\frac{2\sqrt{a}}{\sqrt{a}+\sqrt{b}}+\frac{\sqrt{b}}{\sqrt{b}-\sqrt{a}}\)
\(=\left(\frac{4\sqrt{ab}+\left(\sqrt{a}-\sqrt{b}\right)^2}{2\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\right).\frac{2\sqrt{a}}{\sqrt{a}+\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)
\(=\left(\frac{4\sqrt{ab}+a-2\sqrt{ab}+b}{2\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\right).\frac{2\sqrt{a}}{\sqrt{a}+\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)
\(=\left(\frac{\left(\sqrt{a}+\sqrt{b}\right)^2}{2\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}\right).\frac{2\sqrt{a}}{\sqrt{a}+\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)
\(=\frac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}-\sqrt{b}}\)
\(=\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}=1\)
tick cho mình nha
RÚT GỌN:
\(\frac{2}{\sqrt{ab}}:\left(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\right)^2+\frac{a+b}{a+2\sqrt{ab}+\sqrt{b}}\)