Question

If 2 to the power x is equal to 3 Y is equal to 6 to the power z then prove that 1 by X + 1 by Y is equal to 1 by Z

Answer 1

If $2^{X}=3^{Y}=6^{Z}$ then $\frac{1}{X}+\frac{1}{Y}$ is equal to $\frac{1}{Z}$

What is an exponent?

⇒An exponent refers to the number of times a number is multiplied by itself.

Multiplication of exponents

⇒ It is also known as the law on indices. For multiplication with the same base, we add the powers.

For the same base  $a^{m} \times a^{n}=a^{m+n}$

For the different base and same power  $a^{m} \times b^{m} = (a\times b)^{m}$

Calculation:

⇒ We have been given that $2^{X}=3^{Y}=6^{Z}$ and by this relation, we have to prove that: $\frac{1}{X}+\frac{1}{Y}$= $\frac{1}{Y}$

⇒ $2^{X}=3^{Y}=6^{Z}$

$2^{X}=6^{Z}$

$(2)^{X}=(2 \times3)^{Z}$

$2^{X}=3^{Z}\times 2^{Z}$

By taking power of Y both sides

⇒ $(2^{X})^{Y} =(2^{Z})^{Y} \times(3^{Z})^{Y}$

$(2^{X})^{Y} =(2^{Z})^{Y} \times(3^{Y})^{Z}$

$(2^{X})^{Y} =(2^{Z})^{Y} \times(2^{X})^{Z}$

$2^{XY} =2^Z^Y \times2^X^Z$

XY=ZY+XZ

On dividing by 1/XYZ into both sides we will get

⇒ $\frac{1}{Z}=\frac{1}{X}+\frac{1}{Y}$

Hence, it is now proved that if $2^{X}=3^{Y}=6^{Z}$ then$\frac{1}{X}+\frac{1}{Y}$ is equal to $\frac{1}{Z}$

Learn more about the law of indices here :

brainly.com/question/170984

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