Answer :
If [tex]2^{X}=3^{Y}=6^{Z}[/tex] then [tex]\frac{1}{X}+\frac{1}{Y}[/tex] is equal to [tex]\frac{1}{Z}[/tex]
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 [tex]a^{m} \times a^{n}=a^{m+n}[/tex]
For the different base and same power [tex]a^{m} \times b^{m} = (a\times b)^{m}[/tex]
Calculation:
⇒ We have been given that [tex]2^{X}=3^{Y}=6^{Z}[/tex] and by this relation, we have to prove that: [tex]\frac{1}{X}+\frac{1}{Y}[/tex]= [tex]\frac{1}{Y}[/tex]
⇒ [tex]2^{X}=3^{Y}=6^{Z}[/tex]
[tex]2^{X}=6^{Z}[/tex]
[tex](2)^{X}=(2 \times3)^{Z}[/tex]
[tex]2^{X}=3^{Z}\times 2^{Z}[/tex]
By taking power of Y both sides
⇒ [tex](2^{X})^{Y} =(2^{Z})^{Y} \times(3^{Z})^{Y}[/tex]
[tex](2^{X})^{Y} =(2^{Z})^{Y} \times(3^{Y})^{Z}[/tex]
[tex](2^{X})^{Y} =(2^{Z})^{Y} \times(2^{X})^{Z}[/tex]
[tex]2^{XY} =2^Z^Y \times2^X^Z[/tex]
XY=ZY+XZ
On dividing by 1/XYZ into both sides we will get
⇒ [tex]\frac{1}{Z}=\frac{1}{X}+\frac{1}{Y}[/tex]
Hence, it is now proved that if [tex]2^{X}=3^{Y}=6^{Z}[/tex] then[tex]\frac{1}{X}+\frac{1}{Y}[/tex] is equal to [tex]\frac{1}{Z}[/tex]
Learn more about the law of indices here :
brainly.com/question/170984
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