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Chapter 10: Problem 88

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\sqrt[3]{x y^{2} z} \sqrt{x^{3} y z^{2}}$$

Short Answer

Expert verified
\(\sqrt[3]{x^{5/2} y^{5/2} z^{2}}\)

Step by step solution

01

- Simplify each radical separately

Simplify \(\sqrt[3]{x y^{2} z}\) and \(\sqrt{x^{3} y z^{2}}\) independently.For \(\sqrt[3]{xy^{2}z}\), it remains the same since it's already in simplest form for cube root.For \(\sqrt{x^{3} y z^{2}}\), identify any perfect squares: \(\sqrt{(x^{3})}(\sqrt{y})(\sqrt{z^{2}})\).Here, \(\sqrt{x^{3}} = x^{3/2}\) (since \(x^{3/2} = x^{1.5}\)), \(\sqrt{y} = y^{1/2}\), and \(\sqrt{z^{2}} = z\). Thus, \(\sqrt{x^{3} y z^{2}} = x^{3/2} y^{1/2} z\).
02

- Combine the simplified radicals under one radical

To multiply the simplified radicals, we combine them under a single radical: \(\sqrt[3]{x y^{2} z} \times \sqrt{x^{3/2} y^{1/2} z}\).This equals \(\sqrt[3]{x y^{2} z \cdot x^{3/2} y^{1/2} z} = \sqrt[3]{x^{1 + 3/2} y^{2 + 1/2} z^{1 + 1}} = \sqrt[3]{x^{5/2} y^{5/2} z^{2}}\).
03

- Simplify inside the radical

Combine the exponents inside the radical: \(\sqrt[3]{x^{5/2} y^{5/2} z^{2}}\).Recognize each term is already simplified and no further factorization is possible.

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Key Concepts

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

cube root
The cube root is the reverse operation of cubing a number. When you cube a number, you multiply it by itself three times. For example, since \(2^3 = 8\), the cube root of 8 is 2, written as \(\root 3 \relax{8} = 2\).
Cube roots are useful for simplifying expressions involving radicals.
When simplifying expressions like \(\root 3 \relax {xy^2z}\), we look for values that are cubed to make the simplification process easier. Since \(xy^2z\) doesn鈥檛 have obvious cube roots, it remains as is.
perfect squares
Perfect squares are numbers or expressions that can be written as the square of another number. For example, \(16 = 4^2\) and \(y^4 = (y^2)^2\).
Recognizing these helps simplify square roots.
In our example, \(\root 2 \relax{(x^3 y z^2)}\), we look for perfect squares inside the radical. For \(x^3\), we write it as \(x^{2} \times x\), giving us the square root of a perfect square part.
Thus, \(\root 2 \relax{x^{3} y z^{2}} = \root 2 \relax{x^{3/2}} \times \root 2 \relax{y} \times \root 2 \relax{z^2}\), leading us to simplified \(x^{3/2} y^{1/2} z\).
multiplying radicals
Multiplying radicals involves combining multiple radical expressions under a single radical.
To multiply \(\root 3 \relax{xy^2z}\) and \(\root 2 \relax{x^{3/2} y^{1/2} z}\), we use properties of exponents.
We first separate the radicals, simplify each part, and then multiply inside one radical.
Here鈥檚 how it looks: \(\root 3 \relax{xy^2z} \times \root 2 \relax{x^{3/2} y^{1/2} z}\). This simplifies inside the radical to \( \root 3 \relax{x^{5/2} y^{5/2} z^2}\).
Combined factors are then worked on to form simplified expressions.

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Most popular questions from this chapter

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt{10} \sqrt{14} $$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ 3 \sqrt{2 x^{5}} \cdot 4 \sqrt{10 x^{2}} $$

Find a simplified form for \(f(x) .\) Assumex \(\geq 0\). $$f(x)=\sqrt[4]{x^{5}-x^{4}}+3 \sqrt[4]{x^{9}-x^{8}}$$

If the perimeter of a regular hexagon is \(120 \mathrm{ft}\), what is its area?

f(x) and g(x) are as given. Find \((f \cdot g)(x) \cdot\) Assume that all variables represent non negativereal numbers. $$f(x)=x+\sqrt{7}, g(x)=x-\sqrt{7} $$
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