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Chapter 5: Problem 29

Use properties of exponents to simplify each expression. First express the answer in exponential form. Then evaluate the expression. \(\frac{2^{3}}{2^{7}}\)

Short Answer

Expert verified
The simplified form of the expression is \(\frac{1}{16}\).

Step by step solution

01

Apply the Quotient of Powers Property

Using the quotient of powers property, simplify the expression \(\frac{2^{3}}{2^{7}}\) to become \(2^{3-7}\).
02

Simplify the Exponent

Evaluate the exponent \(3-7\) to become \(-4\), so the expression turns into \(2^{-4}\).
03

Convert Negative Exponent to Positive

A negative exponent can be converted to a positive exponent by taking the reciprocal of the base number. Therefore, \(2^{-4}\) becomes \(\frac{1}{2^{4}}\).
04

Evaluate the Expression

Evaluate \(\frac{1}{2^{4}}\). \(2^{4}\) equals to 16, hence \(\frac{1}{2^{4}} = \frac{1}{16}\).

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Most popular questions from this chapter

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(15,30,60,120, \ldots\)

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 \(-7\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{7}\), when \(a_{1}=4, r=2\)

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. \(a_{1}=-70, d=-5\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{7}\), when \(a_{1}=5, r=-2\).
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