please can anybody solve this question.It is really urgent please.
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please can anybody solve this question.It is really urgent please.
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Solution:
Given to prove:
[tex]\tt\longrightarrow \dfrac{aceg}{bdfh}=\dfrac{a^4+c^4+e^4+g^4}{b^4+d^4+f^4+h^4}[/tex]
Where:
[tex]\tt\longrightarrow \dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=\dfrac{g}{h}[/tex]
And the values b, d, f, h ≠ 0
Let us assume that:
[tex]\tt\longrightarrow \dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}=\dfrac{g}{h}=k[/tex]
Where k is some real number, k ≠ 0
Then we can say:
[tex]\tt\longrightarrow a=bk[/tex]
[tex]\tt\longrightarrow c=dk[/tex]
[tex]\tt\longrightarrow e=fk[/tex]
[tex]\tt\longrightarrow g=hk[/tex]
Now consider Left Hand Side, we have:
[tex]\tt=\dfrac{aceg}{bdfh}[/tex]
Can be written as:
[tex]\tt=\dfrac{(bk)\cdot(dk)\cdot(fk)\cdot(hk)}{bdfh}[/tex]
[tex]\tt=\dfrac{k^4\cdot bdfh}{bdfh}[/tex]
[tex]\tt=k^4[/tex]
Now consider Right Hand Side, we have:
[tex]\tt=\dfrac{a^4+c^4+e^4+g^4}{b^4+d^4+f^4+h^4}[/tex]
[tex]\tt=\dfrac{(bk)^4+(dk)^4+(fk)^4+(hk)^4}{b^4+d^4+f^4+h^4}[/tex]
[tex]\tt=\dfrac{k^4(b^4+d^4+f^4+h^4)}{b^4+d^4+f^4+h^4}[/tex]
[tex]\tt=k^4[/tex]
We observe that LHS = RHS
Therefore:
[tex]\tt\longrightarrow \dfrac{aceg}{bdfh}=\dfrac{a^4+c^4+e^4+g^4}{b^4+d^4+f^4+h^4}[/tex]
Hence Proved..!!
Learn More:
If a : b and c : d are two ratios such that a : b : : c : d. Then the following results hold true.
1. Invertendo.
[tex]\tt\longrightarrow b : a : : d : c[/tex]
2. Alternendo.
[tex]\tt\longrightarrow a : c : : b : d[/tex]
3. Componendo.
[tex]\tt\longrightarrow (a + b) : b: : (c + d) : d[/tex]
4. Dividendo.
[tex]\tt\longrightarrow (a - b) : b: : (c - d) : d[/tex]
5. Componendo and dividendo.
[tex]\tt\longrightarrow(a + b) : (a - b): : (c + d) :(c - d)[/tex]
6. Convertendo.
[tex] \tt\longrightarrow a : (a - b): :c:(c - d)[/tex]