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Chapter 2: Problem 12

The numbers are too large to be handled by a calculator. These exercises require an understanding of the concepts. Write \(5^{3} \cdot 25^{2000}\) as a power of \(5 .\)

Short Answer

Expert verified
The short answer for this question is: The expression \(5^3 \cdot 25^{2000}\) can be written as a power of \(5\) as \(5^{4003}\).

Step by step solution

01

Rewrite the expression in terms of powers of 5

We rewrite the expression \(25^{2000}\) in terms of power of \(5\), by replacing \(25\) with \(5^2\): \[5^3 \cdot (5^2)^{2000}\]
02

Simplify the expression using power of a power property

In this step, we'll use the power of a power property which states that \((a^b)^c = a^{b\cdot c}\). Applying this property to our expression, we get: \[5^3 \cdot 5^{(2 \cdot 2000)}\]
03

Evaluating the expression

Now it's time to evaluate the expression: \[5^3 \cdot 5^{(2 \cdot 2000)} = 5^3 \cdot 5^{4000}\]
04

Combine the exponents using the product of powers property

We'll use the product of powers property which states that \(a^b \cdot a^c = a^{b+c}\). Applying this property to our expression, we get: \[5^3 \cdot 5^{4000} = 5^{(3 + 4000)}\]
05

Simplify the expression

Finally, we simplify the expression: \[5^{(3 + 4000)} = 5^{4003}\] The expression \(5^3 \cdot 25^{2000}\) can be written as a power of \(5\) as \(5^{4003}\).

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Most popular questions from this chapter

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

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{6}-4 x^{2}+5}{x^{2}-3 x+1} $$

Suppose \(t(x)=\frac{5}{4 x^{3}+3}\). (a) Show that the point (-1,-5) is on the graph of \(t\) (b) Give an estimate for the slope of a line containing (-1,-5) and a point on the graph of \(t\) very close to (-1,-5)

Explain why the composition of two rational functions is a rational function.

Suppose \(s(x)=\frac{x^{2}+2}{2 x-1}\) (a) Show that the point (1,3) is on the graph of \(s\). (b) Show that the slope of a line containing (1,3) and a point on the graph of \(s\) very close to (1,3) is approximately -4 [Hint: Use the result of Exercise \(25 .]\)
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