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Chapter 7: Problem 4

Multiply, if possible. Then simplify. $$ \sqrt[4]{8} \cdot \sqrt[4]{32} $$

Short Answer

Expert verified
The result of multiplying \(\sqrt[4]{8}\) and \(\sqrt[4]{32}\) and then simplifying is 4.

Step by step solution

01

Multiply radicals

Both radical expressions have the same index (4), hence we can multiply \(\sqrt[4]{8}\) and \(\sqrt[4]{32}\) inside the fourth root, as per the properties of radicals. This results in \(\sqrt[4]{8 \cdot 32}\).
02

Calculate the product inside the radical

The product inside the fourth root is \(8 \cdot 32 = 256\). This gives us \(\sqrt[4]{256}\).
03

Simplify the fourth root

Given that \(4^4 = 256\), the fourth root of 256 simplifies to 4. Hence, \(\sqrt[4]{256} = 4\).

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Key Concepts

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

Properties of Radicals
Radicals can be a bit confusing, but knowing their properties can simplify many problems. One important property is the ability to multiply radicals with the same index. If you have two radicals, say \(\sqrt[n]{a}\) and \(\sqrt[n]{b}\), you can combine them as \(\sqrt[n]{ab}\) so long as the index (\(n\)) is the same. This means you multiply the numbers inside the radicals and keep them under a single root.
For instance, with the fourth roots \(\sqrt[4]{8}\) and \(\sqrt[4]{32}\), since both have an index of 4, you can multiply the numbers inside them. This property makes the process more efficient and leads to simpler expressions.
Simplifying Radicals
Once you've combined the numbers inside a radical, the next step is to simplify it. Simplifying involves breaking down the number to find its perfect roots or reducing it to its simplest form.A simplified radical has no perfect powers left inside. For example, after multiplying, the expression becomes \(\sqrt[4]{256}\). Recognize that 256 is actually \(2^8\), which can be further simplified given that \(4^2 = 16\) and \(16^2 = 256\).When simplifying, look for numbers whose powers match the radical's index. This indicates that the number can result in a whole number when taken out from under the radical.
Fourth Root
The fourth root of a number is essentially asking what number, when multiplied by itself four times, equals the original number. For example, finding the fourth root of 256 means asking what number gives 256 when raised to the power of 4. In this instance, that number is 4, because \(4^4 = 256\).To find the fourth root, check whether the number inside the radical is a perfect fourth power. If it is, the fourth root will be a whole number. If not, it suggests further simplification is needed. Recognizing powers, especially higher ones like cubes and fourths, helps solve these problems more quickly and accurately.

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Most popular questions from this chapter

How is the graph of \(y=\sqrt{x}-5\) translated from the graph of \(y=\sqrt{x} ?\) F. shifted 5 units left G. shifted 5 units right H. shifted 5 units up J. shifted 5 units down

The graph of \(y=-\sqrt{x}\) is shifted 4 units up and 3 units right. Which equation represents the new graph? A. \(y=-\sqrt{x-4}+3\) B. \(y=-\sqrt{x-3}+4\) C. \(y=-\sqrt{x+3}+4\) D. \(y=-\sqrt{x+4}+3\)

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(2 \sqrt{x+4}=3 \sqrt{x-1}\)

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