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Chapter 12: Problem 124

$$\text { Divide and simplify: } \frac{\sqrt[3]{40 x^{2} y^{6}}}{\sqrt[3]{5 x y}}$$

Short Answer

Expert verified
The simplified form of the expression \(\frac{\sqrt[3]{40 x^{2} y^{6}}}{\sqrt[3]{5 x y}}\) is \(2xy\).

Step by step solution

01

- Simplify the Cube Roots

Firstly, break down the cube root of each term in the numerator and denominator. The cube root of \(40x^{2}y^{6}\) can be written as \(\sqrt[3]{8x^{2}y^{3}} \times \sqrt[3]{5y^{3}}\) and the cube root of \(5xy\) can be written as \(\sqrt[3]{125y^{3}}\).
02

- Further Simplify the Cube Roots

Notice that certain parts in the cube roots can be simplified. \(\sqrt[3]{8x^{2}y^{3}}\) simplifies to \(2xy\), and \(\sqrt[3]{5y^{3}}\) simplifies to \(5y\). Similarly, \(\sqrt[3]{125y^{3}}\) simplifies to \(5y\). Replacing the simplifiable parts in the equation, the given expression then becomes \(\frac{2xy \times 5y}{5y}\).
03

- Cancel out Common Terms

In the simplified expression \(\frac{2xy \times 5y}{5y}\), the term \(5y\) is common both in the numerator and denominator. So, they can be cancelled out, resulting in the final simplified form: \(2xy\).

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Key Concepts

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

Cube Roots
Cube roots are similar to square roots, but instead of finding a number that, when multiplied by itself twice (squared), gives you the original number, you find a number that, when multiplied by itself twice and then once more (cubed), results in the original number. For example, the cube root of 8 is 2 because if we multiply 2 by itself three times (\(2 \times 2 \times 2\)), we get 8.
Cube roots are often denoted using the radical symbol with a small three, like this: \(\sqrt[3]{8}\). Calculating cube roots can sometimes be tricky, especially when dealing with expressions that involve variables.
  • Understand that finding the cube root of an expression means finding a factor that, when cubed, gives back the original expression.
  • Always break down complex expressions into factors that can individually be cube rooted.
  • Practice simplifying numbers under the cube root by looking for perfect cubes 鈥� numbers like 8, 27, 64, which have whole number cube roots.
Simplifying Expressions
Simplifying expressions is all about making them easier to work with, without changing their meaning. In algebra, this often involves reducing fractions, factoring, and eliminating any unnecessary terms.
To simplify expressions that include cube roots, first, express the number or the term under the cube root as a product of the biggest perfect cube you can find and another factor. For instance, in \(\sqrt[3]{40x^{2}y^{6}}\), you can factor 40 as \(8 \times 5\), where 8 is a perfect cube.
  • Reduce the expression by performing cube roots on the perfect cube parts.
  • Look for common terms in expressions so they can be canceled out in the fraction form.
  • Combine like terms to shrink the expression further.
Factoring
Factoring involves breaking down a complex expression into simpler parts or smaller expressions that, when multiplied together, give you the original expression. In the context of cube roots and simplifying expressions, factoring helps separate parts that can be independently simplified.
For example, when you encounter \(\sqrt[3]{40x^{2}y^{6}}\), recognition that 40 can be broken down into its factors allows you to take the cube root systematically.
  • Always identify the largest perfect cubes within a term 鈥� they simplify naturally and help condense the expression.
  • Factoring expressions reduces complexity and helps in identifying common terms in numerators and denominators.
  • Master common factoring methods like grouping and using identities like difference of cubes.

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Most popular questions from this chapter

Solve each equation. $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$

Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

In parts (a)-(c), graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to _______.

a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$

Simplify: \(\left(-2 x^{3} y^{-2}\right)^{-4}\)
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