Question

The value of $\left(a + \frac{1}{b}\right)^r \left(a - \frac{1}{b}\right)^s \div \left(b + \frac{1}{a}\right)^r \left(b - \frac{1}{a}\right)^s$ is:

• A. $(\frac{a}{b})^{r+s}$
• B. $(\frac{b}{a})^{r+s}$
• C. $({a}{b})^{r+s}$
• D. $(\frac{a^r}{b^s})$

Answer 1

Answer:

$(\frac{a}{b})^{r+s}$

The simplified value of the given expression is $\left(\frac{a}{b}\right)^{r+s}$.

The given expression is:
$\frac{\left(a + \frac{1}{b}\right)^r \left(a - \frac{1}{b}\right)^s}{\left(b + \frac{1}{a}\right)^r \left(b - \frac{1}{a}\right)^s}$

1. Simplify the fractional terms inside each bracket:

   • $\left(a + \frac{1}{b}\right) = \frac{ab + 1}{b}$

   • $\left(a - \frac{1}{b}\right) = \frac{ab - 1}{b}$

   • $\left(b + \frac{1}{a}\right) = \frac{ab + 1}{a}$

   • $\left(b - \frac{1}{a}\right) = \frac{ab - 1}{a}$

2. Substitute these back into the expression:
   $\frac{\left(\frac{ab + 1}{b}\right)^r \cdot \left(\frac{ab - 1}{b}\right)^s}{\left(\frac{ab + 1}{a}\right)^r \cdot \left(\frac{ab - 1}{a}\right)^s}$

3. Separate the numerators and denominators using exponent rules:
   $\frac{\frac{(ab + 1)^r}{b^r} \cdot \frac{(ab - 1)^s}{b^s}}{\frac{(ab + 1)^r}{a^r} \cdot \frac{(ab - 1)^s}{a^s}}$

4. Cancel out the common terms $(ab + 1)^r$ and $(ab - 1)^s$ from the top and bottom:
   $\frac{\frac{1}{b^r \cdot b^s}}{\frac{1}{a^r \cdot a^s}} = \frac{a^r \cdot a^s}{b^r \cdot b^s}$

5. Combine using the law of indices ($x^m \cdot x^n = x^{m+n}$):
   $\frac{a^{r+s}}{b^{r+s}} = \mathbf{\left(\frac{a}{b}\right)^{r+s}}$