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Chapter 10: Problem 21

Multiply. $$ \sqrt[5]{x-2} \sqrt[5]{(x-2)^{2}} $$

Short Answer

Expert verified
\(\root 5{(x-2)^{3}}\)

Step by step solution

01

- Understand the Expression

Recognize that we are multiplying two radical expressions with the same index of 5, which are \(\root 5{x-2} \) and \(\root 5{(x-2)^{2}}\).
02

- Use the Property of Radicals

Use the property of radicals that states \(\root[n]{a} \times \root[n]{b} = \root[n]{ab}\). In our case, both radicals have an index of 5, so we can combine them into a single radical: \(\root 5{(x-2) \times (x-2)^{2}}\).
03

- Simplify the Expression Inside the Radical

Simplify the expression inside the radical. Multiply the terms inside: \((x-2) \times (x-2)^{2} = (x-2)^{1} \times (x-2)^{2} = (x-2)^3\).
04

- Write the Simplified Radical Expression

Combine the simplified expression back under the same radical: \(\root 5{(x-2)^{3}}\).

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Key Concepts

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

Radical Multiplication
In math, you often encounter radical expressions. Multiplying these radicals follows specific rules. Always start by recognizing the index of the radicals. The index is the small number above the root symbol, indicating how many times a number must be multiplied by itself to return the initial value. When multiplying radicals with the same index, you can combine them under a single radical. For example, multiplying \(\root5{x-2} \times \root5{(x-2)^{2}}\) uses the property \(\root[n]{a} \times \root[n]{b} = \root[n]{ab}\). This means you can simplify to \(\root5{(x-2) \times (x-2)^{2}}\). Remember to ensure both radicals have the same index before combining.
Index of Radicals
The index of a radical tells you the degree of the root. If there's no number, it implies a square root (index 2). For example, \(\root3{8}\) is a cube root (index 3). When working with radicals, knowing the index is crucial. For multiplication, the radicals must share the same index to combine. In our exercise, both radicals \(\root 5{x-2}\) and \(\root 5{(x-2)^{2}}\) have an index of 5. This similarity allows combining them easily into one: \(\root 5{(x-2) \times (x-2)^{2}}\). Differing indices can't be combined directly; they require conversion to a common index first.
Simplifying Radical Expressions
Simplifying radicals involves reducing the expression under the root to its simplest form. Start by combining the terms inside. From \(\root 5{(x-2) \times (x-2)^{2}}\), multiply the bases with exponents: \((x-2) \times (x-2)^{2}=(x-2)^1 \times (x-2)^{2} = (x-2)^{3}\). The new expression is \(\root 5{(x-2)^{3}}\). Always aim to express the contents in simplest terms. Even higher power expressions like \((a^2 \times a^3)\) can be combined: \((a^{2+3}) = a^{5}\). Practice recognizing these patterns and using them to simplify any radical step-by-step, ensuring the final expression is as concise as possible.

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Most popular questions from this chapter

The absolute value of a complex number \(a+b i\) is its distance from the origin. Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}} .\) Find the absolute value of each complex number. $$ |-1+i| $$

Each side of a regular octagon has length \(s .\) Find a formula for the distance \(d\) between the parallel sides of the octagon.

Find the midpoint of each segment with the given endpoints. \((-3.4,8.1)\) and \((4.8,-8.1)\)

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[4]{(5+3 x)^{3}}}{\sqrt[3]{(5+3 x)^{2}}}$$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{5 a^{2}} \sqrt[3]{2 a} $$
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