Problem 4 Without using a calculator or co... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 3: Problem 4

Without using a calculator or computer, determine which of the two numbers $2^{400}$ and $17^{100}$ is larger.

Short Answer

Expert verified

$17^{100}$ is larger than $2^{400}$.

Step by step solution

Rewrite 17 as a power of 2

Since we want to compare the numbers, we're going to rewrite $17^{100}$ as a power of 2. We know that $17 = 2^4 + 1 = 2^4 + 2^0$. Now let's raise this expression to the power of 100.

Use binomial theorem to expand the expression

We will use the binomial theorem to expand $(2^4 + 2^0)^{100}$. The binomial theorem states that for any positive integer $n$ and real numbers $a$ and $b$: $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$ Here, we have $a = 2^4$ and $b = 2^0$. Applying the binomial theorem to our expression: $(2^4 + 2^0)^{100} = \sum_{k=0}^{100} \binom{100}{k} (2^4)^{100-k}(2^0)^k$

Simplify the sum

Let's simplify the sum we got in step 2: $(2^4 + 2^0)^{100} = \sum_{k=0}^{100} \binom{100}{k} 2^{400-4k}2^k = \sum_{k=0}^{100} \binom{100}{k} 2^{400-3k}$

Rewrite the expression with $2^{400}$

We are trying to compare $2^{400}$ and $17^{100}$, so let's rewrite the expression with $2^{400}$ included. We are specifically interested in the term with $2^{400}$. In the sum, this occurs when $k = 0$, and we get the following term: $\binom{100}{0} 2^{400} = 2^{400}$. Now we rewrite the sum as: $(2^4 + 2^0)^{100} = 2^{400} + \sum_{k=1}^{100} \binom{100}{k} 2^{400-3k}$

Compare the numbers

Now, let's compare the two numbers: $2^{400}$ and $(2^4 + 2^0)^{100} = 2^{400} + \sum_{k=1}^{100} \binom{100}{k} 2^{400-3k}$ From the above expression, we can see that $(2^4 + 2^0)^{100}$ has an additional sum of positive terms after $2^{400}$. Therefore, the number $(2^4 +2^0)^{100} = 17^{100}$ is larger compared to $2^{400}$.

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91影视

Without using a calculator or computer, determine which of the two numbers \(2^{400}\) and \(17^{100}\) is larger.

Short Answer

Step by step solution

Rewrite 17 as a power of 2

Use binomial theorem to expand the expression

Simplify the sum

Rewrite the expression with \(2^{400}\)

Compare the numbers

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