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Chapter 2: Problem 11

The numbers are too large to be handled by a calculator. These exercises require an understanding of the concepts. Write \(2^{5} \cdot 8^{1000}\) as a power of 2 .

Short Answer

Expert verified
The short answer is: \(2^{5} \cdot 8^{1000} = 2^{3005}\).

Step by step solution

01

Write 8 as a power of 2

First, we need to write \(8\) as a power of 2 because we are asked to express the given number as a power of 2. Since \(8 = 2^3\), we can rewrite \(8^{1000}\) as \((2^3)^{1000}\). Now our expression becomes: \(2^5 \cdot (2^3)^{1000}\)
02

Apply the power rule

The power rule states that for any real numbers a, m, and n, we have \((a^m)^n = a^{mn}\). Applying this rule to our expression, we have: \(2^5 \cdot 2^{(3 \cdot 1000)}\)
03

Apply the product rule

The product rule states that for any real numbers a, m, and n, we have \(a^m \cdot a^n = a^{m+n}\). Applying this rule to our expression, we get: \(2^{(5 + 3 \cdot 1000)}\)
04

Simplify the exponent

Finally, we just need to simplify the exponent: \(2^{(5 + 3000)} = 2^{3005}\) The expression \(2^{5} \cdot 8^{1000}\) as a power of 2 is: \(2^{3005}\)

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Most popular questions from this chapter

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r s)(x) $$

Suppose \(t\) is a zero of the polynomial \(p\) defined by $$ p(x)=3 x^{5}+7 x^{4}+2 x+6 $$ Show that \(\frac{1}{t}\) is a zero of the polynomial \(q\) defined by $$ q(x)=3+7 x+2 x^{4}+6 x^{5} $$.

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{4 x-5}{x+7} $$

A bicycle company finds that its average cost per bicycle for producing \(n\) thousand bicycles is \(a(n)\) dollars, where $$ a(n)=800 \frac{3 n^{2}+n+40}{16 n^{2}+2 n+45} $$ What will be the approximate cost per bicycle when the company is producing many bicycles?

Write the indicated expression as \(a\) polynomial. $$ (p(x))^{2} s(x) $$
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