a: A=(-1)+(-1)+...+(-1)+2023

=2023-1011=1012

\(\sqrt{5+\sqrt{21}}-\sqrt{5-\sqrt{21}}\\ =\dfrac{\left(\sqrt{10+2\sqrt{21}}-\sqrt{10-2\sqrt{21}}\right)}{\sqrt{2}}\\ =\dfrac{\left(\sqrt{7+2\sqrt{7}.\sqrt{3}+3}-\sqrt{7-2\sqrt{7}.\sqrt{3}+3}\right)}{\sqrt{2}}\\ =\dfrac{\sqrt{7}+\sqrt{3}-\sqrt{7}+\sqrt{3}}{\sqrt{2}}=\dfrac{2\sqrt{3}}{\sqrt{2}}=\sqrt{6}\)

\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{10+2\sqrt{21}}-\sqrt{10-2\sqrt{21}}\right)\)

\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{7}+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}\right)\)

\(=\dfrac{1}{\sqrt{2}}\left(\left|\sqrt{7}+\sqrt{3}\right|-\left|\sqrt{7}-\sqrt{3}\right|\right)\)

\(=\dfrac{2\sqrt{3}}{\sqrt{2}}=\sqrt{6}\)

21/21=1/1

3/3=1/1

4/5=4/5 phân số tối giản ùi 5/6 cx thế 

ủng hộ nhiệt tình nha

21/21=1,3/3=1

Hai cái còn lại là ps tối giản rùi

Lời giải:
Đặt biểu thức là $A$. Ta có:

\(A=(5+\sqrt{21})(\sqrt{7}-\sqrt{3}).\sqrt{2}.\sqrt{5-\sqrt{21}}\)

\(=(5+\sqrt{21})(\sqrt{7}-\sqrt{3}).\sqrt{10-2\sqrt{21}}\)

\(=(5+\sqrt{21})(\sqrt{7}-\sqrt{3}).\sqrt{(\sqrt{7}-\sqrt{3})^2}\)

\(=(5+\sqrt{21})(\sqrt{7}-\sqrt{3})|\sqrt{7}-\sqrt{3}|=(5+\sqrt{21})(\sqrt{7}-\sqrt{3})^2\)

\(=(5+\sqrt{21})(10-2\sqrt{21})=2(5+\sqrt{21})(5-\sqrt{21})=2(5^2-21)=8\)

Ta có: \(\left(5+\sqrt{21}\right)\cdot\left(\sqrt{14}-\sqrt{6}\right)\cdot\sqrt{5-\sqrt{21}}\)

\(=\dfrac{\left(10+2\sqrt{21}\right)\cdot\left(\sqrt{7}-\sqrt{3}\right)\cdot\sqrt{10-2\sqrt{21}}}{2}\)

\(=\dfrac{\left(\sqrt{7}+\sqrt{3}\right)^2\cdot\left(\sqrt{7}-\sqrt{3}\right)^2}{2}\)

=8

\(\sqrt{5-\sqrt{21}}+\sqrt{5+\sqrt{21}}\)

\(=\sqrt{\left(\sqrt{\frac{7}{2}}-\sqrt{\frac{3}{2}}\right)^2}+\sqrt{\left(\sqrt{\frac{7}{2}}+\sqrt{\frac{3}{2}}\right)^2}\)

\(=\left|\sqrt{\frac{7}{2}}-\sqrt{\frac{3}{2}}\right|+\left|\sqrt{\frac{7}{2}}+\sqrt{\frac{3}{2}}\right|\)

\(=\sqrt{\frac{7}{2}}-\sqrt{\frac{3}{2}}+\sqrt{\frac{7}{2}}+\sqrt{\frac{3}{2}}\)

\(=2.\sqrt{\frac{7}{2}}\)

\(=\sqrt{14}\)

Chúc bạn học gỏi và tíck cho mìk vs nha!

\(\sqrt{\sqrt{5}-\sqrt{5-\sqrt{21-4\sqrt{5}}}}\)

\(=\sqrt{\sqrt{5}-\sqrt{5-\sqrt{\sqrt{20^2}-2.\sqrt{20}+1}}}\)

\(=\sqrt{\sqrt{5}-\sqrt{5-\sqrt{\left(\sqrt{20}-1\right)^2}}}\)

\(=\sqrt{\sqrt{5}-\sqrt{5-\left|\sqrt{20}-1\right|}}\)

\(=\sqrt{\sqrt{5}-\sqrt{5-\sqrt{20}+1}}\)

\(=\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}\)

\(=\sqrt{\sqrt{5}-\sqrt{\sqrt{5^2}-2\sqrt{5}+1}}\)

\(=\sqrt{\sqrt{5}-\sqrt{\left(\sqrt{5}-1\right)^2}}\)

\(=\sqrt{\sqrt{5}-\left|\sqrt{5}-1\right|}\)

\(=\sqrt{\sqrt{5}-\sqrt{5}+1}\)

\(=1\)

\(=\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}\)

\(=\sqrt{\sqrt{5}-\sqrt{5}+1}=1\)

Tham khảo ạ

ở cái dấu = thứ 2 tại sao lại nhân với \(\dfrac{1}{\sqrt{2}}\) ạ?

a: Số số hạng của dãy số là:

(2023-1):1+1=2023-1+1=2023(số)

A=1-2+3-4+...+2021-2022+2023

=(1-2)+(3-4)+...+(2021-2022)+2023

=(-1)+(-1)+...+(-1)+2023

\(=-\frac{2022}{2}+2023=2023-1011=1012\)

b:

Số số hạng của dãy số là:

(313-1):3+1=312:3+1=104+1=105(số)

\(B=1-4+7-10+\cdots+307-310+313\)

=(1-4)+(7-10)+...+(307-310)+313

=(-3)+(-3)+...+(-3)+313

\(=\left(-3\right)\cdot\frac{104}{2}+313\)

=313-156

=157

c: \(C=-2194\cdot21952195+2195\cdot21942194\)

\(=2194\cdot2195\left(-10001+10001\right)\)

=0

\(\sqrt{21-8\sqrt{5}}\)\(-\sqrt{21-4\sqrt{5}}\)

\(=\sqrt{16-2.4\sqrt{5}+5}\)\(-\sqrt{20-2\sqrt{20}+1}\)

\(=\sqrt{\left(4-\sqrt{5}\right)^2}\)\(-\sqrt{\left(\sqrt{20}-1\right)}\)

\(=4-\sqrt{5}-\left(\sqrt{20}-1\right)\)

\(=4-\sqrt{5}-\sqrt{20}+1\)

\(=5-\sqrt{5}-2\sqrt{5}\)

\(=5-3\sqrt{5}\)