(b) $$ \frac{2^{2 n} \cdot 2^{n} \cdot 2^{n}-8^{n}}{2^{5} \cdot 2^{3 n}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2^{2n} \cdot 2^{n} \cdot 2^{n} - 8^{n}}{2^{5} \cdot 2^{3n}}
$$

**Step 1: Simplify the Numerator**

First, combine the exponents for the product of powers of 2:

$$
2^{2n} \cdot 2^{n} \cdot 2^{n} = 2^{2n + n + n} = 2^{4n}
$$

Next, simplify $$8^{n}$$:

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

So, the numerator becomes:

$$
2^{4n} - 2^{3n}
$$

**Step 2: Simplify the Denominator**

Combine the exponents in the denominator:

$$
2^{5} \cdot 2^{3n} = 2^{5 + 3n}
$$

**Step 3: Combine Numerator and Denominator**

The expression now is:

$$
\frac{2^{4n} - 2^{3n}}{2^{5 + 3n}}
$$

Factor out $$2^{3n}$$ from the numerator:

$$
\frac{2^{3n}(2^{n} - 1)}{2^{5 + 3n}}
$$

Cancel out $$2^{3n}$$ from the numerator and denominator:

$$
\frac{2^{n} - 1}{2^{5}} = \frac{2^{n} - 1}{32}
$$

**Final Simplified Form:**

$$
\frac{2^{n} - 1}{32}
$$

**Answer:** After simplifying, the expression is (2ⁿ – 1) divided by 32. Thus,

(2ⁿ − 1)/32
The simplified form of the expression is $$ \frac{2^{n} - 1}{32} $$.

(b)