Evaluate a x - a y z - a y - a z x

Evaluate (a^x/a^y)× (a^y/a^z)^x =(a^xz/a^yz)× (a^yx/a^zx) #160; #160; ∵ (a^x)^z=a^xz =a^yz/a^yz=(a^x/a^z)^y ...

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Standard VII

Mathematics

Power of a Power

Evaluate a...

Question

Evaluate ${(\frac{a^{x}}{a^{y}})}^{z} \times {(\frac{a^{y}}{a^{z}})}^{x}$

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Solution

Evaluate
$(\frac{a^{x}}{a^{y}}) \times (\frac{a^{y}}{a^{z}})^{x}$
$= (\frac{a^{x z}}{a^{y z}}) \times (\frac{a^{y x}}{a^{z x}})$ $∵ (a^{x})^{z} = a^{x z}$
$= \frac{a^{y z}}{a^{y z}} = (\frac{a^{x}}{a^{z}})^{y}$

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Similar questions

Q. If

\frac{a x + b y}{x + y} = \frac{b x + a z}{x + z} = \frac{a y + b z}{y + z}

and x + y + z

\neq

0 then show that each ratio is

\frac{a + b}{2} .

Q. Evaluate:

a^{x - y} * b^{y - z} * a^{y - x} * b^{z - y} * 2^{5}

A_{x y}, A_{y z}, A_{z x}

be the area of projections of an area

A

on the

x y, y z

abd

z x

planes respectively, then considering

\to A

, a vector quantity whose direction in normal to surface

A

and its component along the coordinate axes be

A_{x}, A_{y}

and

A_{z}

such that

\to A = A_{x} i + A_{y} j + A_{z} k

, therefore we can say

| A_{x} | = A_{z y}, ∣ ∣ A_{y} ∣ ∣ = A_{x z}, | A_{z} | = A_{x y}

and hence the area

∣ ∣ ∣ \to A ∣ ∣ ∣ = \sqrt{{A_{x}}^{2} + {A_{y}}^{2} + {A_{z}}^{2}} \Rightarrow A^{2} = {A_{x y}}^{2} + {A_{y z}}^{2} + {A_{z x}}^{2}

The area of

△

whose vertices are

(1, 2, 3), (2, 4, 1), (3, 4, 5)

respectively is

A_{x y}, A_{y z}, A_{z x}

be the area of projections of an area

A

on the

x y, y z

abd

z x

planes respectively, then considering

\to A

, a vector quantity whose direction in normal to surface

A

and its component along the coordinate axes be

A_{x}, A_{y}

and

A_{z}

such that

\to A = A_{x} i + A_{y} j + A_{z} k

, therefore we can say

| A_{x} | = A_{z y}, ∣ ∣ A_{y} ∣ ∣ = A_{x z}, | A_{z} | = A_{x y}

and hence the area

∣ ∣ ∣ \to A ∣ ∣ ∣ = \sqrt{{A_{x}}^{2} + {A_{y}}^{2} + {A_{z}}^{2}} \Rightarrow A^{2} = {A_{x y}}^{2} + {A_{y z}}^{2} + {A_{z x}}^{2}

A plane makes intercepts

a, b, c

x, y, z

axes respectively, then twice the area of

△ A B C

including both its sides

(a, b, c

are +ve

)

A_{x y}, A_{y z}, A_{z x}

be the area of projections of an area

A

on the

x y, y z

abd

z x

planes respectively, then considering

\to A

, a vector quantity whose direction in normal to surface

A

and its component along the coordinate axes be

A_{x}, A_{y}

and

A_{z}

such that

\to A = A_{x} i + A_{y} j + A_{z} k

, therefore we can say

| A_{x} | = A_{z y}, ∣ ∣ A_{y} ∣ ∣ = A_{x z}, | A_{z} | = A_{x y}

and hence the area

∣ ∣ ∣ \to A ∣ ∣ ∣ = \sqrt{{A_{x}}^{2} + {A_{y}}^{2} + {A_{z}}^{2}} \Rightarrow A^{2} = {A_{x y}}^{2} + {A_{y z}}^{2} + {A_{z x}}^{2}

Through a point

P (h, k, l)

a plane is drawn at right angle to

O P

to meet the coordinate axes in

A, B, C

and

O P = p,

then the area of

△ A B C

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Evaluate (axay)z×(ayaz)x

Evaluate(axay)×(ayaz)x=(axzayz)×(ayxazx) ∵(ax)z=axz=ayzayz=(axaz)y

Evaluate ${(\frac{a^{x}}{a^{y}})}^{z} \times {(\frac{a^{y}}{a^{z}})}^{x}$

Evaluate
$(\frac{a^{x}}{a^{y}}) \times (\frac{a^{y}}{a^{z}})^{x}$
$= (\frac{a^{x z}}{a^{y z}}) \times (\frac{a^{y x}}{a^{z x}})$ $∵ (a^{x})^{z} = a^{x z}$
$= \frac{a^{y z}}{a^{y z}} = (\frac{a^{x}}{a^{z}})^{y}$