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Chapter 8: Problem 160

Simplify. Assume all variables are positive (a) \(\frac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}$$\left(\frac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)

Short Answer

Expert verified
The simplified expression is \(3 a^3 b c^{-1}\).

Step by step solution

01

- Simplify the numerator

Combine the exponents in the numerator using the property of exponents: \ \ \(a^{\frac{3}{4}} \times a^{-\frac{1}{4}} = a^{\frac{3}{4} - \frac{1}{4}} = a^{\frac{2}{4}} = a^{\frac{1}{2}}\ Thus, the numerator is now: \ a^{\frac{1}{2}}\)
02

- Simplify the fraction

Simplify the fraction using the property of exponents, \(\frac{a^m}{a^n} = a^{m-n}\): \ \ \(\frac{a^{\frac{1}{2}}}{a^{-\frac{10}{4}}} = a^{\frac{1}{2} - (-\frac{10}{4})} = a^{\frac{1}{2} + \frac{10}{4}} = a^{\frac{1}{2} + \frac{5}{2}} = a^{\frac{6}{2}} = a^3\)
03

- Simplify inside the parentheses

Combine the exponents inside the parentheses separately for \(b\) and \(c\): \ \ \(\frac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}} = 27 b^{\frac{2}{3} - (-\frac{7}{3})} c^{-\frac{5}{2} - \frac{1}{2}} = 27 b^{\frac{2}{3} + \frac{7}{3}} c^{-\frac{6}{2}} = 27 b^{3} c^{-3}\)
04

- Apply the exponent outside the parentheses

Raise the entire expression inside the parentheses to the power of \(\frac{1}{3}\): \ \ \[\left(27 b^3 c^{-3}\right)^{\frac{1}{3}} = 27^{\frac{1}{3}} b^{3 \cdot \frac{1}{3}} c^{-3 \cdot \frac{1}{3}} = 3 b c^{-1}\]
05

- Combine all simplified terms

Combine the results from the previous steps: \ \ \(a^3 \times 3 b c^{-1} = 3 a^3 b c^{-1}\)

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Key Concepts

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

simplification techniques
Simplifying algebraic expressions requires a methodical approach to combine like terms and reduce complexity. Here are some techniques to simplify expressions:
  • Combine Like Terms: Merge terms with the same variable and exponent. For example, \(a^2 + a^2 = 2a^2\).
  • Use the Properties of Exponents: Apply the properties of exponents to simplify expressions. For example, \(x^2 \times x^3 = x^{2+3} = x^5\).
  • Simplify Fractions: When dealing with fractions, simplify by canceling common factors in the numerator and denominator. For instance, \(\frac{b^4}{b^2} = b^{4-2} = b^2\).
  • Simplify Inside Parentheses: Before applying an outer exponent, simplify the expression inside the parentheses. For example, \(\left(a^2 b^3\right)^2 = a^{2\cdot2} b^{3\cdot2} = a^4 b^6\).
Applying these techniques consistently leads to a clearer, more simplified final expression.

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Most popular questions from this chapter

Explain the difference of finding the \(n^{\text {th }}\) root of a number when the index is even compared to when the index is odd.

Why is there no real number equal to \(\sqrt{-64}\) ?

Simplify. Assume all variables are positive (a) \(\frac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}\) \(\left(\frac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{-\frac{1}{2}}}\right)^{\frac{1}{4}}\)

Explain why \(\sqrt[4]{-64}\) is not a real number but \(\sqrt[3]{-64}\) is.

Simplify using absolute value signs as needed. (a) \(\sqrt{n^{21}}\) (b) \(\sqrt[3]{q^{8}} \)(c)\( \sqrt[8]{n^{10}}\)
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