a) $$ \frac{8^{n} q^{n}}{2^{3 n}-3^{n}} $$

Question

UpStudy AI · Accepted Answer

Certainly! Let's simplify the expression you've provided:

$$
\frac{8^{n} q^{n}}{2^{3n} - 3^{n}}
$$

### Step 1: Simplify the Numerator

First, recognize that $$8^n$$ can be rewritten in terms of powers of 2:

$$
8^n = (2^3)^n = 2^{3n}
$$

So, the numerator becomes:

$$
8^{n} q^{n} = 2^{3n} q^{n}
$$

### Step 2: Analyze the Denominator

The denominator is already expressed as:

$$
2^{3n} - 3^{n}
$$

### Step 3: Rewrite the Entire Expression

Now, substitute the simplified numerator back into the original expression:

$$
\frac{2^{3n} q^{n}}{2^{3n} - 3^{n}}
$$

### Step 4: Factor Out $$2^{3n}$$ from the Denominator

To further simplify, factor $$2^{3n}$$ out of the denominator:

$$
2^{3n} - 3^{n} = 2^{3n} \left(1 - \frac{3^{n}}{2^{3n}}\right) = 2^{3n} \left(1 - \left(\frac{3}{8}\right)^n\right)
$$

### Step 5: Simplify the Fraction

Now, substitute this back into the expression:

$$
\frac{2^{3n} q^{n}}{2^{3n} \left(1 - \left(\frac{3}{8}\right)^n\right)} = \frac{q^{n}}{1 - \left(\frac{3}{8}\right)^n}
$$

### Final Simplified Form

$$
\frac{q^{n}}{1 - \left(\frac{3}{8}\right)^n}
$$

### Interpretation

This simplified form expresses the original fraction in terms of $$q^n$$ and a geometric factor $$\left(\frac{3}{8}\right)^n$$. This can be particularly useful for analyzing the behavior of the expression as $$n$$ grows large or for further algebraic manipulations, depending on the context of the problem.

If you have more context or further parts to the problem, feel free to share them, and I can assist you further!
$$ \frac{q^{n}}{1 - \left(\frac{3}{8}\right)^n} $$

a)

Upstudy AI Solution

Answer

Solution

Step 1: Simplify the Numerator

Step 2: Analyze the Denominator

Step 3: Rewrite the Entire Expression

Step 4: Factor Out from the Denominator

Step 5: Simplify the Fraction

Final Simplified Form

Interpretation

Mind Expander

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