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Chapter 5: Problem 37

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. \(\frac{2 x^{5} \cdot 3 x}{15 x^{6}}\)

Short Answer

Expert verified
The simplified form of the expression is \(\frac{2}{5}\).

Step by step solution

01

Simplify Numerical Coefficients

Multiply and divide the numerical coefficients. \(\frac{2*3}{15} = \frac{6}{15}\) can be simplified to \(\frac{2}{5}\)
02

Simplify Exponential Terms

Combine the powers of x. According to the rule of exponents, when we multiply expressions with the same base, we add the exponents: \(x^{5} * x = x^{5+1} = x^{6}\)
03

Divide Exponential Terms

Now, divide the exponential term in the numerator by the exponential term in the denominator. \(\frac{x^{6}}{x^{6}} = x^{6-6} = x^{0}\) According to the properties of exponents, anything raised to the power of 0 is 1.
04

Combine the Results

Combine the results of numerical simplification and power simplification. So, \(\frac{2}{5} * 1 = \frac{2}{5}\)

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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 a sequence is geometric, we can write as many terms as we want by repeatedly multiplying by the common ratio.

Use the appropriate formula shown above to find \(2+4+6+8+\cdots+200\), the sum of the first 100 positive even integers.

What is a sequence? Give an example with your description.

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{8}\), when \(a_{1}=40,000, r=0.1\).

Enough curiosities involving the Fibonacci sequence exist to warrant a flourishing Fibonacci Association. It publishes a quarterly journal. Do some research on the Fibonacci sequence by consulting the research department of your library or the Internet, and find one property that interests you. After doing this research, get together with your group to share these intriguing properties.
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