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Chapter 10: Problem 84

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\sqrt[5]{a^{3} b} \sqrt{a b}$$

Short Answer

Expert verified
\(a^{11/10} b^{7/10}\)

Step by step solution

01

Express the radicals with fractional exponents

Rewrite each radical expression using fractional exponents. The fifth root can be written as \(\root{5}{a^3 b} = (a^3 b)^{1/5}\), and the square root can be written as \(\root{2}{a b} = (ab)^{1/2}\).
02

Combine the exponents

Using the property of exponents that \((x^m)(x^n) = x^{m+n}\), combine the terms together. This results in: \((a^3 b)^{1/5} (a b)^{1/2} = a^{3/5} b^{1/5} \cdot a^{1/2} b^{1/2}\).
03

Add the exponents of like bases

When multiplying expressions with the same base, add their exponents. Thus, we get: \((a^{3/5} a^{1/2}) (b^{1/5} b^{1/2})\). Add the exponents for \(a\) and \(b\) separately. So: \((a^{3/5 + 1/2}) (b^{1/5 + 1/2})\). Convert fractions to have a common denominator. This gives: \((a^{6/10 + 5/10}) (b^{2/10 + 5/10})\).
04

Simplify the exponents

Simplify the exponent addition to get: \((a^{11/10}) (b^{7/10})\).
05

Rewrite using radicals

Express the solution back in radical form if necessary. Since we started with radicals, we can represent the final expression as: \(\root{10}{a^{11}} \root{10}{b^7}\).

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Key Concepts

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

fractional exponents
Fractional exponents are another way to express roots. They provide a powerful tool for simplifying complex expressions. Instead of writing \(\root{n}{x}\), we write it using exponents as \(x^{1/n}\).
For example:
  • \(x^{1/2}\) means the square root of \(x\)
  • \(x^{1/3}\) means the cube root of \(x\)
  • \(x^{m/n}\) means the n-th root of \(x^m\)
Using fractional exponents allows us to apply the properties of exponents easily, simplifying expressions like \(\sqrt[5]{a^3 b} \sqrt{a b}\) more straightforwardly.
properties of exponents
Understanding the properties of exponents is crucial for simplifying expressions. Here's an overview of the most important properties:
  • Product of Powers: \(x^m \cdot x^n = x^{m+n}\)
  • Quotient of Powers: \(x^m \/ x^n = x^{m-n}\)
  • Power of a Power: \((x^m)^n = x^{m\cdot n}\)
  • Power of a Product: \( (xy)^m = x^m \cdot y^m \)
  • Negative Exponent: \(x^{-m} = 1/x^m\)
  • Zero Exponent: \(x^0 = 1\)
These properties make it possible to simplify expressions like \sqrt[5]{a^3 b} \sqrt{a b} \ by converting roots to fractional exponents and then combining them.
combining like terms
Combining like terms is a fundamental technique when simplifying algebraic expressions. Like terms have the same variables raised to the same powers. Here are a few guidelines:
  • Add or subtract the coefficients of like terms.
    Example: \(3a + 4a = 7a \).
  • Only combine terms with the same variable and exponent.
    Example: \(2a^2 + 3a \) can't be combined.
  • When dealing with exponents, remember to add the exponents if the bases are the same.
    Example: \(a^{1/2} \cdot a^{3/5} = a^{1/2 + 3/5}\).
In our exercise, combining like terms helps us turn \(a^{3/5} \cdot a^{1/2} = a^{3/5 + 1/2}\) into a simplified single term.
radical expressions
Radical expressions involve roots of numbers or variables, like square roots or cube roots. They can be challenging to simplify. Here are the main steps to manage them:
  • Express radicals using fractional exponents.
    Example: \(\sqrt[3]{x^2}\) becomes \(x^{2/3}\).
  • Use properties of exponents to simplify.
    Example: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} = (ab)^{1/2}\).
  • Combine like terms if possible.
    Example: \(\sqrt{a^2b} = |a| \sqrt{b} \).
In the exercise, rewriting \sqrt[5]{a^3 b} \sqrt{a b}\ helped convert it to fractional exponents, making it easier to apply the properties of exponents for simplification.

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Most popular questions from this chapter

Find a simplified form for \(f(x) .\) Assumex \(\geq 0\). $$f(x)=\sqrt{x^{3}-x^{2}}+\sqrt{9 x^{3}-9 x^{2}}-\sqrt{4 x^{3}-4 x^{2}}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$(r-\sqrt[4]{s^{3}})(3 r-\sqrt[5]{s})$$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[5]{a^{3}(b-c)^{4}} \sqrt[5]{a^{7}(b-c)^{4}} $$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{s^{2} t^{4}} \sqrt[3]{s^{4} t^{6}} $$

Simplify. $$ \frac{5-\sqrt{5 i}}{\sqrt{5 i}} $$
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