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Chapter 2: Problem 13

The numbers are too large to be handled by a calculator. These exercises require an understanding of the concepts. Write \(2^{100} \cdot 4^{200} \cdot 8^{300}\) as a power of 2 .

Short Answer

Expert verified
The short answer is that the expression \(2^{100} \cdot 4^{200} \cdot 8^{300}\) simplifies to \(2^{1400}\).

Step by step solution

01

Write the given expression as a power of 2

We have the expression \(2^{100} \cdot 4^{200} \cdot 8^{300}\). Write the bases 4 and 8 as powers of 2. We know that \(4 = 2^2\) and \(8 = 2^3\).
02

Replace the bases

Replace the bases 4 and 8 in the expression with their equivalent expressions as powers of 2 from Step 1: \[ (2^{100}) \cdot (2^2)^{200} \cdot (2^3)^{300}. \]
03

Use the power of a power rule

Apply the power of a power rule, which states that raising an exponent to another exponent means multiplying the exponents: \[ (2^{100}) \cdot (2^{2 \cdot 200}) \cdot (2^{3 \cdot 300}) \]
04

Simplify the exponents

Multiply the exponents: \[ (2^{100}) \cdot (2^{400}) \cdot (2^{900}) \]
05

Use the product rule for exponents with the same base

Apply the product rule for exponents, which states that when we multiply numbers with the same base, we add their exponents: \[ 2^{100 + 400 + 900} \]
06

Calculate the sum

Add the exponents together: \[ 2^{1400} \]
07

Final answer

The result of \(2^{100} \cdot 4^{200} \cdot 8^{300}\) is \(2^{1400}\).

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