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Chapter 5: Problem 136

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(5^{2} \cdot 5^{-2}>2^{5} \cdot 2^{-5}\)

Short Answer

Expert verified
The given statement is false. The corrected statement to make it true is \(5^{2} \cdot 5^{-2} = 2^{5} \cdot 2^{-5}\).

Step by step solution

01

Understand the Laws of Exponents

The law of exponents states that \(a^n \cdot a^m = a^{n+m}\). In this case, \(5^{2} \cdot 5^{-2}\) becomes \(5^{2+(-2)}\) and \(2^{5} \cdot 2^{-5}\) becomes \(2^{5+(-5)}\).
02

Simplify the Expression

Applying the law of exponents to the expression, we get \(5^{2+(-2)} > 2^{5+(-5)}\). This simplifies to \(5^{0} > 2^{0}\). Any number (except zero) raised to the power of zero equals 1. Therefore, the inequality simplifies to \(1 > 1\)
03

Determine Whether the Statement is True or False

\(1 > 1\) is obviously false, because 1 is not greater than 1. Therefore, the given statement is false.
04

Make the Necessary Changes to Produce a True Statement

To make the statement true, the inequality needs to change from greater than to equal to. So, the corrected statement becomes \(5^{2} \cdot 5^{-2} = 2^{5} \cdot 2^{-5}\)

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the first term of an arithmetic sequence is 5 and the third term is \(-3\), then the fourth term is \(-7\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(-9,-5,-1,3, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2000, r=-1\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sequence \(1,4,8,13,19,26, \ldots\) is an arithmetic sequence.

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. \(a_{1}=-70, d=-5\)
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