Math Problem Statement

110110000010110_2 A. 27x2^(10)+22 B. 27×2^(10)+ 44 C. 27×2^(11) +22 D.27×2^(11)+44

Solution

To solve this problem, we need to interpret the binary number 1101100000101102110110000010110_2 and convert it into its decimal (base-10) equivalent.

The binary number 1101100000101102110110000010110_2 can be expanded as follows: 1×214+1×213+0×212+1×211+1×210+0×29+0×28+0×27+0×26+1×25+0×24+1×23+1×22+0×21+0×201 \times 2^{14} + 1 \times 2^{13} + 0 \times 2^{12} + 1 \times 2^{11} + 1 \times 2^{10} + 0 \times 2^9 + 0 \times 2^8 + 0 \times 2^7 + 0 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0

Calculating powers of 2: 214=16384,213=8192,211=2048,210=1024,25=32,23=8,22=42^{14} = 16384, \quad 2^{13} = 8192, \quad 2^{11} = 2048, \quad 2^{10} = 1024, \quad 2^5 = 32, \quad 2^3 = 8, \quad 2^2 = 4

Adding these together: 16384+8192+2048+1024+32+8+4=2769216384 + 8192 + 2048 + 1024 + 32 + 8 + 4 = 27692

Now, comparing this with the given options:

  • Option A: 27×210+22=27×1024+22=27648+22=2767027 \times 2^{10} + 22 = 27 \times 1024 + 22 = 27648 + 22 = 27670
  • Option B: 27×210+44=27×1024+44=27648+44=2769227 \times 2^{10} + 44 = 27 \times 1024 + 44 = 27648 + 44 = 27692
  • Option C: 27×211+22=27×2048+22=55296+22=5531827 \times 2^{11} + 22 = 27 \times 2048 + 22 = 55296 + 22 = 55318
  • Option D: 27×211+44=27×2048+44=55296+44=5534027 \times 2^{11} + 44 = 27 \times 2048 + 44 = 55296 + 44 = 55340

The correct match is Option B: 27×210+44=2769227 \times 2^{10} + 44 = 27692.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

  1. How do you convert a binary number to decimal manually?
  2. What are the common uses of binary numbers in computing?
  3. How can powers of two be computed quickly?
  4. What is the significance of base-2 in digital systems?
  5. How does understanding binary assist in understanding machine code?

Tip: When working with binary numbers, remember each digit represents an increasing power of 2, starting from the rightmost digit (which is 202^0).

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Math Problem Analysis

Mathematical Concepts

Binary to Decimal Conversion
Powers of 2
Algebraic Expressions

Formulas

Binary to Decimal Conversion: \(1 \times 2^n + 0 \times 2^{n-1} + \dots + 1 \times 2^0\)
Powers of 2
Algebraic Expressions: a × 2^n + b

Theorems

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Suitable Grade Level

Grades 10-12