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Chapter 8: Problem 59

Simplify. $$\sqrt{27 a^{5} b^{9}}$$

Short Answer

Expert verified
The simplified form is \[3a^2b^4 \sqrt{3ab}.\]

Step by step solution

01

- Factor Inside the Radical

Factor the expression inside the radical sign into its prime factors. It simplifies into \(27 = 3^3\), so rewrite it as \(3^3 \) and break down \(a^5\) and \(b^9 \) to make the exponents clearer: \[\sqrt{3^3 \, a^5 \, b^9}\].
02

- Apply the Square Root to Each Factor

Apply the square root rule \(\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}\) individually to each factor: \[\sqrt{3^3} \, \cdot \, \sqrt{a^5} \, \cdot \, \sqrt{b^9}.\]
03

- Simplify the Square Roots of Variables with Exponents

Simplify each term inside the square roots: \[\sqrt{3^3} = 3^{3/2} \, and \, \sqrt{a^5} = a^{5/2}, \, and \, \sqrt{b^9} = b^{9/2}.\]
04

- Rewrite Exponents

Recall that any fractional exponent \(x^{m/n}\) can be written as the root \(\sqrt[n]{x^m}\). In simpler exponential terms, we get: \[3^{3/2} = (3^2)^{1/2} \, \cdot 3. Thus, 3^{3/2} \, = 3 \sqrt{3}, a^{5/2} = \sqrt{a^5} = a^2 \sqrt{a}, and \, b^{9/2} = \sqrt{b^9} = b^4 \sqrt{b}.\]
05

- Combine Simplified Terms

Put the simplified radical terms together: \[3 \sqrt{3} \cdot a^2 \sqrt{a} \cdot b^4 \sqrt{b}.\]
06

- Simplify Further

Combine terms outside the radicals and terms inside the radicals: \[3 \cdot a^2 \cdot b^4 \sqrt{3ab}.\]

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Key Concepts

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

Prime Factorization
Prime factorization is a method used to break down a number into its prime components. This is very useful when simplifying radical expressions. To perform a prime factorization, you continually divide the number by the smallest prime (starting from 2, 3, 5, etc.) until you can't anymore.
Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 * 3 = 9.
Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, instead of writing 3 * 3 * 3, you can write 3^3.
Simplifying Radicals
Simplifying radicals involves expressing the radical in its simplest form. This can be done using prime factorization, applying square roots to each factor, and combining like terms.

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