To simplify the expression [tex]\(\frac{2^2 \times 2^3 \times 2^4}{2 \times 2^5}\)[/tex], follow these steps:
1. Combine the exponents in the numerator:
- The numerator is [tex]\(2^2 \times 2^3 \times 2^4\)[/tex].
- When multiplying powers with the same base, you add the exponents: [tex]\(2^2 \times 2^3 \times 2^4 = 2^{2+3+4} = 2^9\)[/tex].
2. Combine the exponents in the denominator:
- The denominator is [tex]\(2 \times 2^5\)[/tex].
- Remember that [tex]\(2\)[/tex] can be written as [tex]\(2^1\)[/tex], so the denominator is [tex]\(2^1 \times 2^5\)[/tex].
- When multiplying powers with the same base, you add the exponents: [tex]\(2^1 \times 2^5 = 2^{1+5} = 2^6\)[/tex].
3. Simplify the fraction by subtracting the exponents in the numerator and denominator:
- You have [tex]\(\frac{2^9}{2^6}\)[/tex].
- When dividing powers with the same base, you subtract the exponents: [tex]\(2^9 \div 2^6 = 2^{9-6} = 2^3\)[/tex].
4. Calculate the simplified value:
- [tex]\(2^3\)[/tex] means [tex]\(2 \times 2 \times 2\)[/tex].
- [tex]\(2 \times 2 = 4\)[/tex].
- [tex]\(4 \times 2 = 8\)[/tex].
So, the simplified expression is:
[tex]\[
\frac{2^2 \times 2^3 \times 2^4}{2 \times 2^5} = 2^3 = 8
\][/tex]
Therefore, the step-by-step solution leads to the final simplified value of [tex]\(8\)[/tex].
