Question

Somebody help me? I don't understand how it is solved (the factor thing is like saying x100 times)

Answer 1

Answer:

$\sf E=\dfrac{x^2 \cdot x^4 \cdot x^6 \cdot x^8 ... 100\:factors}{x \cdot x^3 \cdot x^5 \cdot x^7 ... 100\:factors}=x^{100}$

Step-by-step explanation:

Given:

$\sf E=\dfrac{x^2 \cdot x^4 \cdot x^6 \cdot x^8 ...}{x \cdot x^3 \cdot x^5 \cdot x^7 ... }$

As:

$\sf E=\dfrac{x^2 \cdot x^4 \cdot x^6 \cdot x^8 \cdot ... }{x \cdot x^3 \cdot x^5 \cdot x^7 \cdot ... }=\dfrac{x^2}{x^1} \cdot \dfrac{x^4}{x^3} \cdot \dfrac{x^6}{x^5} \cdot \dfrac{x^8}{x^7} \cdot ...}$

Apply the exponent rule  $\sf \dfrac{a^b}{a^c}=a^{b-c}$

$\begin{aligned}\sf \implies \dfrac{x^2}{x^1} \cdot \dfrac{x^4}{x^3} \cdot \dfrac{x^6}{x^5} \cdot \dfrac{x^8}{x^7} \cdot... & = \sf x^{(2-1)} \cdot x^{(4-3)} \cdot x^{(6-5)} \cdot x^{(8-7)} \cdot ...\\ & = \sf x^1 \cdot x^1 \cdot x^1 \cdot x^1 \cdot ...\end{aligned}$

As there are 100 factors, then $\sf x^1$ is multiplied by itself 100 times ⇒ $\sf x^{100}$

Therefore:

$\sf E=\dfrac{x^2 \cdot x^4 \cdot x^6 \cdot x^8 ... 100\:factors}{x \cdot x^3 \cdot x^5 \cdot x^7 ... 100\:factors}=x^{100}$

Answer 2

sum of first n even nos =n(n+1)

sum of first n odd nos=n²

So

Simplify the numerator

• x²x⁴x⁶..=x^(2+4+6..100times )

Simplify the denominator

• xx³x⁵..=x^{1+3+5..100times)

Sum of first 100 even nos

• 100(100-1)=100(99)=9900

Sum of first 100 odd nos

• 100²=10000.

So

The equation yields as

• E=x⁹⁹⁰⁰/x¹⁰⁰⁰⁰
• E=1/x¹⁰⁰