Let's break down the problem step by step.

Part (a)

The given expression is: $(1 - 2x)^{\frac{3}{5}} = 1 + ax + bx^2 + \dots$ You are asked to find the values of $a$ and $b$.

This is essentially a binomial expansion problem where we need to expand $(1 - 2x)^{\frac{3}{5}}$ using the generalized binomial theorem for fractional powers. The binomial theorem for any power $n$ can be written as:

$(1 + u)^n = 1 + n u + \frac{n(n-1)}{2!} u^2 + \frac{n(n-1)(n-2)}{3!} u^3 + \dots$

In this case, $u = -2x$ and $n = \frac{3}{5}$.

So, the expansion starts as: $(1 - 2x)^{\frac{3}{5}} = 1 + \frac{3}{5}(-2x) + \frac{\frac{3}{5}\left(\frac{3}{5} - 1\right)}{2!}(-2x)^2 + \dots$

Simplifying the first two terms:

• The first term is just 1.
• The second term is: $\frac{3}{5}(-2x) = -\frac{6}{5}x$

For the third term (coefficient of $x^2$):

• The general term for $x^2$ is: $\frac{\frac{3}{5}\left(\frac{3}{5} - 1\right)}{2!}(-2x)^2 = \frac{\frac{3}{5}\left(-\frac{2}{5}\right)}{2} \cdot 4x^2 = \frac{-6}{25} \cdot 2x^2 = -\frac{12}{25}x^2$

Thus, the expansion up to $x^2$ is: $(1 - 2x)^{\frac{3}{5}} = 1 - \frac{6}{5}x - \frac{12}{25}x^2 + \dots$

From this, we can compare it with $1 + ax + bx^2 + \dots$, giving:

• $a = -\frac{6}{5}$
• $b = -\frac{12}{25}$

Part (b)

Now, you are asked to use the values of $a$ and $b$ to find $30^{\frac{3}{5}}$ in the form of $\frac{m}{n}$.

We can use the binomial approximation for $30^{\frac{3}{5}}$. Notice that: $30 = 32 - 2$ So, we write: $30^{\frac{3}{5}} = (32 - 2)^{\frac{3}{5}} = 32^{\frac{3}{5}} \left(1 - \frac{2}{32}\right)^{\frac{3}{5}}$

We can approximate the binomial expansion as: $(1 - \frac{1}{16})^{\frac{3}{5}} \approx 1 + \left(\frac{3}{5}\right) \left(-\frac{1}{16}\right) + \dots = 1 - \frac{3}{80}$

Thus, we have: $30^{\frac{3}{5}} \approx 32^{\frac{3}{5}} \left(1 - \frac{3}{80}\right)$

Next, calculate $32^{\frac{3}{5}}$: $32^{\frac{3}{5}} = 2^3 = 8$

Therefore: $30^{\frac{3}{5}} \approx 8 \times \left(1 - \frac{3}{80}\right) = 8 \times \frac{77}{80} = \frac{616}{80} = \frac{77}{10}$

So, $m = 77$ and $n = 10$.

Final Answers:

1. $a = -\frac{6}{5}$, $b = -\frac{12}{25}$
2. $30^{\frac{3}{5}} = \frac{77}{10}$

Would you like a more detailed explanation on any step, or is there another part you'd like to explore further?

Related Questions:

1. What is the binomial expansion for $(1 + x)^n$ for any real number $n$?
2. How do you find higher-order terms in a binomial expansion with fractional powers?
3. How can you approximate roots using binomial expansions?
4. What happens to the accuracy of approximations as more terms are added in a binomial expansion?
5. Can the binomial theorem be applied to other functions beyond polynomials?

Tip:

When expanding binomials with fractional exponents, always check the radius of convergence to ensure the approximation is valid for the values of $x$ used.