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Chapter 7: Problem 39

Simplify the products. Give exact answers. $$\sqrt[4]{9} \cdot \sqrt[4]{27}$$

Short Answer

Expert verified
The simplified product is \(3^{5/4}\).

Step by step solution

01

- Write the expressions in exponential form

Rewrite the fourth roots as exponents. We have \ \(\sqrt[4]{9} = 9^{1/4}\) and \(\sqrt[4]{27} = 27^{1/4}\).
02

- Multiply the exponents

Recall that when multiplying like bases, you can add the exponents of the terms: \ \(9^{1/4} \times 27^{1/4} = (9 \times 27)^{1/4}\).
03

- Simplify the product inside the parentheses

Find the product of 9 and 27: \ \(9 \times 27 = 243\). Therefore, the expression becomes \(243^{1/4}\).
04

- Simplify the fourth root

Rewrite 243 as a power of a smaller number: \ \(243 = 3^5\). Therefore, \(243^{1/4} = (3^5)^{1/4}\).
05

- Apply the power rule

Use the power rule \((a^m)^n = a^{m \times n}\) to simplify: \ \((3^5)^{1/4} = 3^{5 \times 1/4} = 3^{5/4}\).

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Key Concepts

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

simplifying expressions
Simplifying expressions can make a complex problem much more manageable. For instance, when simplifying expressions with exponents and radicals, rewriting them in a more familiar form can help.
Often, we see roots and exponents. To simplify, rewrite roots as exponents. For example: \(\text{\textbackslash sqrt[4]\textbraceleft 9\textbraceright\textbackslash = 9\textasciicircum{1/4}}\). This change makes it easier to apply other rules. Remember, simplification is about making expressions easier to handle!
product of roots
The product of roots is often encountered in mathematics. If you multiply roots with the same index, you can combine them.
For example: \(\text{\textbackslash sqrt[4]\textbraceleft 9\textbraceright \textbackslash& \textbackslash sqrt[4] \textbraceleft 27\textbraceright = (9 \times 27)\textasciicircum{1/4}}\). This makes it easier to continue simplifying since you're working with single expressions.
This concept shows how combining terms can lead to simpler calculations.
exponential form
Converting expressions into exponential form is crucial for simplifying radicals and roots. Instead of working directly with roots, use the exponent equivalent.
For instance, \(\text{\textbackslash sqrt[4]\textbraceleft a\textbraceright = a\textasciicircum{1/4}}\). Recognizing this form allows you to use the properties of exponents to simplify further.
This method is particularly helpful when dealing with higher-index roots.
power rule
The power rule is key when simplifying expressions with exponents. It states: \((a\textasciicircum{m})\textasciicircum{n} = a\textasciicircum{(m \times n)}}\). Use this rule to simplify nested exponents.
For instance, from our example: \((3\textasciicircum{5})\textasciicircum{1/4} = 3\textasciicircum{(5 \times 1/4)} = 3\textasciicircum{5/4}}\).

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