Chapter 12: Problem 4

Express cach of the following expressions using a single positive index: (a) \(x^{4} x^{7}\) (b) \(x^{2}(-x)\) (c) \(\frac{x^{2}}{x}\). (d) \(\frac{x^{-2}}{x^{-1}}\) (c) \(\left(x^{-2}\right)^{4}\) (f) \(\left(x^{-25} x^{-3.5}\right)^{2}\)

Short Answer

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Question: Simplify each of the following expressions using a single positive index. a) \(x^{4} x^{7}\) b) \(x^{2}(-x)\) c) \(\frac{x^{2}}{x}\) d) \(\frac{x^{-2}}{x^{-1}}\) e) \(\left(x^{-2}\right)^{4}\) f) \(\left(x^{-25} x^{-3.5}\right)^{2}\) Answer: a) \(x^{11}\) b) \(x\) c) \(x\) d) \(\frac{1}{x}\) e) \(\frac{1}{x^8}\) f) \(\frac{1}{x^{57}}\)

Step by step solution

(a) Simplify using the product rule

To simplify \(x^{4} x^{7}\), we use the product rule: When we multiply two numbers with the same base, we simply add their exponents. In this case, the base is x and the exponents are 4 and 7. So, we have \(x^{4} x^{7} = x^{4+7} = x^{11}\).

(b) Simplify using the product rule

To simplify \(x^{2}(-x)\), first, we notice that \(-x\) is the same as \(x^{-1}\). Using the product rule, we add the exponents: \(x^{2} x^{-1} = x^{2 + (-1)} = x^{1}\). We express the answer as just \(x\), because the exponent 1 doesn't need to be written explicitly.

(c) Simplify using the quotient rule

To simplify \(\frac{x^{2}}{x}\), we use the quotient rule: When we divide two numbers with the same base, we subtract the exponent of the denominator from the exponent of the numerator. In this case, we have \(\frac{x^{2}}{x} = \frac{x^2}{x^1} = x^{2 - 1} = x^{1}\), and we express the answer as \(x\).

(d) Simplify using the quotient rule

To simplify \(\frac{x^{-2}}{x^{-1}}\), we use the quotient rule: \(\frac{x^{-2}}{x^{-1}} = x^{-2 - (-1)} = x^{-2 + 1} = x^{-1}\). Since the goal is to express this with a positive index, we'll rewrite \(x^{-1}\) as \(\frac{1}{x}\).

(e) Simplify using the power rule

To simplify \(\left(x^{-2}\right)^{4}\), we use the power rule: When a number with an exponent is raised to another exponent, we multiply the exponents together. In this case, we have \(\left(x^{-2}\right)^{4} = x^{-2 \cdot 4} = x^{-8}\). To represent the result with a positive index, we rewrite \(x^{-8}\) as \(\frac{1}{x^8}\).

(f) Simplify using the product and power rules

To simplify \(\left(x^{-25} x^{-3.5}\right)^{2}\), first, we simplify the expression inside the parentheses using the product rule: \(x^{-25} x^{-3.5} = x^{-25 + (-3.5)} = x^{-28.5}\). Now, we apply the power rule to \((x^{-28.5})^{2}\): \(x^{-28.5 \cdot 2} = x^{-57}\). To express this with a positive index, we rewrite \(x^{-57}\) as \(\frac{1}{x^{57}}\).

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Express cach of the following expressions using a single positive index: (a) \(x^{4} x^{7}\) (b) \(x^{2}(-x)\) (c) \(\frac{x^{2}}{x}\). (d) \(\frac{x^{-2}}{x^{-1}}\) (c) \(\left(x^{-2}\right)^{4}\) (f) \(\left(x^{-25} x^{-3.5}\right)^{2}\)

Short Answer

Step by step solution

(a) Simplify using the product rule

(b) Simplify using the product rule

(c) Simplify using the quotient rule

(d) Simplify using the quotient rule

(e) Simplify using the power rule

(f) Simplify using the product and power rules

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