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Chapter 5: Problem 33

Use properties of exponents to simplify each expression. Express answers in exponential form with positive exponents only. Assume that any variables in denominators are not equal to zero. \(\frac{\left(x^{3}\right)^{4}}{\left(x^{2}\right)^{7}}\)

Short Answer

Expert verified
The simplified expression in exponential form with positive exponents is \( \frac{1}{x^{2}} \)

Step by step solution

01

Simplify the Numerator

Using the power of a power property, the numerator can be simplified by multiplying the exponents. Thus, \( (x^{3})^{4} = x^{3*4} = x^{12} \).
02

Simplify the Denominator

Similarly, in the denominator, \( (x^{2})^{7} = x^{2*7} = x^{14} \). Thus, the expression becomes \( \frac{x^{12}}{x^{14}} \).
03

Simplify the Expression

When dividing like bases, the exponents are subtracted. Hence, the expression can be simplified as \( x^{12-14} = x^{-2} \). It is negative as the denominator's exponent is larger.
04

Express the Expression with Positive Exponents

The final step is to convert the negative exponent into a positive exponent. According to the properties of exponents, \( x^{-2} \) can be rewritten as \( \frac{1}{x^{2}} \).

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Most popular questions from this chapter

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010. Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,18.4 \%\) of American women ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.6\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American women ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American women ages 25 and older who will be college graduates by \(2019 .\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{30}\), when \(a_{1}=2, r=-1\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{6}\), when \(a_{1}=18, r=-\frac{1}{3}\).

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{2}, r=2\)

The sum, \(S_{n}\), of the first n terms of an arithmetic sequence is given by $$ S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right), $$ in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by $$ S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}, $$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(4,10,16,22, \ldots\)
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