Question

if 2 to power 2x=5to the power 5y=100 to the power 3z then prove 5xy-15yz-6xz=0​

Answer 1

$\textsf{\large{\underline{Correct Question}:}}$

• If 2²ˣ = 5⁵ʸ = 100³ᶻ then prove that 5xy - 30yz - 6xz = 0

$\textsf{\large{\underline{Solution}:}}$

Let us assume that:

$\rm: \longmapsto {2}^{x} = {5}^{5y} = {100}^{3z} = k$

Therefore:

$\rm: \longmapsto {2}^{x} = k$

$\rm: \longmapsto 2 = {k}^{^{1}/_{x}} - (i)$

Similarly, we can write:

$\rm: \longmapsto 5= {k}^{^{1}/_{5y}} - (ii)$

$\rm: \longmapsto 100= {k}^{^{1}/_{3z}} - (iii)$

Again:

$\rm: \longmapsto 100 = {5}^{2} \times {2}^{2}$

From (i), (ii) and (iii), we can write:

$\rm: \longmapsto {k}^{^{1}/_{3z}} = {k}^{^{2}/_{5y}} \times {k}^{^{2}/_{x}}$

$\rm: \longmapsto {k}^{^{1}/_{3z}} = {k}^{^{2}/_{5y} + ^{2}/_{x}}$

Comparing base, we get:

$\rm: \longmapsto \dfrac{1}{3z} = \dfrac{2}{5y} + \dfrac{2}{x}$

$\rm: \longmapsto \dfrac{1}{3z} = \dfrac{2x + 10y}{5xy}$

On cross multiplying, we get:

$\rm: \longmapsto 5xy = 3z(2x + 10y)$

$\rm: \longmapsto 5xy = 6xz + 30yz$

$\rm: \longmapsto 5xy - 30yz - 6xz = 0$

Hence Proved.!!

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Law of Indices: If a, b are positive real numbers and m, n are rational numbers, then the following results hold.

$\rm 1. \: \: {a}^{m} \times {a}^{n} = {a}^{m + n}$

$\rm 2. \: \: ({a}^{m})^{n} = {a}^{mn}$

$\rm 3. \: \: \dfrac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}$

$\rm 4. \: \: {a}^{m} \times {b}^{m} = {(ab)}^{m}$

$\rm5. \: \: \bigg(\dfrac{a}{b} \bigg)^{m} = \dfrac{ {a}^{m} }{ {b}^{m} }$

$\rm6. \: \: {a}^{ - n} = \dfrac{1}{ {a}^{n} }$

$\rm7. \: \: {a}^{n} = {b}^{n} \rightarrow a = b, n \neq0$

$\rm8. \: \: {a}^{m} = {a}^{n} \rightarrow m = n, a \neq 1$