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Chapter 0: Problem 67

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\sqrt{x}\left(\frac{1}{4 x}\right)^{5 / 2}\)

Short Answer

Expert verified
\(\frac{1}{32x^2}\)

Step by step solution

01

Simplify the square root

Convert the square root to an exponent. Recall that \(\frac{1}{2}\) exponent represents the square root. Thus, \(\sqrt{x} = x^{1/2}\).
02

Simplify the fraction

Rewrite the fraction \(\frac{1}{4x}\) with negative exponents. \(\frac{1}{4x}=4^{-1}x^{-1}\).
03

Apply the power rule

Distribute the exponent \(5/2\) to both terms inside the parentheses: \((4^{-1}x^{-1})^{5/2} = 4^{(-1 \cdot 5/2)} x^{(-1 \cdot 5/2)} = 4^{-5/2} x^{-5/2}\).
04

Combine the exponents

Multiply the terms from step 1 and step 3 using the addition rule for exponents: \(x^{1/2} \times 4^{-5/2} x^{-5/2} = 4^{-5/2} x^{1/2 - 5/2} = 4^{-5/2} x^{-4/2} = 4^{-5/2} x^{-2}\).
05

Simplify the expression

Using the rule that negative exponents indicate reciprocals, convert \(4^{-5/2}\) and \(x^{-2}\) to positive exponents: \(4^{-5/2} = 1/4^{5/2}\) and \(x^{-2} = 1/x^2\).
06

Simplify the constants

Convert \(4^{5/2}\) to base value. Knowing \(4 = 2^2\), then \( 2^{2 \times 5/2} = 2^5\), which simplifies to 32. So, \(4^{5/2} = 32\).
07

Put it all together

Combine everything to get \(\frac{1}{32x^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponents
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    = 1/x^n = 1 x^{-n}} }}ewline This is useful when simplifying complex expressions involving negative exponents denoted Algebra:< ewline exponents are: Algebrav.. (meaning ewline standalone for) that of especially usefulrwrite an expression intoewline after to-to- unacdowed from exponentes and ewline) for)and ewline

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