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Chapter 7: Problem 91

Simplify. $$\sqrt[3]{8 x^{4}}+\sqrt[3]{27 x^{4}}$$

Short Answer

Expert verified
5 \sqrt[3]{x^4}

Step by step solution

01

Simplify each cube root individually

First, simplify \(\text{\sqrt[3]{8x^{4}}}\). Since \(\text{8 = 2^3}\), we get \(\text{\sqrt[3]{8x^{4}}} = \sqrt[3]{2^3 x^4}\). This simplifies to 2 \sqrt[3]{x^4}.
02

Simplify the second cube root

Next, simplify \(\text{\sqrt[3]{27x^{4}}}\). Since \(\text{27 = 3^3}\), we get \(\text{\sqrt[3]{27x^{4}}} = \sqrt[3]{3^3 x^4}\). This simplifies to 3 \sqrt[3]{x^4}.
03

Combine the simplified terms

Now, add the simplified terms together: \(\text{2 \sqrt[3]{x^4} + 3 \sqrt[3]{x^4}}\). This gives \(\text{(2 + 3) \sqrt[3]{x^4} = 5 \sqrt[3]{x^4}}\).

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Key Concepts

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

Understanding Cube Roots
A cube root finds a number that, when multiplied by itself three times, equals the original number. For example, \(\text{2^3 = 8}\), so the cube root of 8 is 2. Cube roots help simplify expressions involving constants and variables raised to powers.
For our example, \(\text{\sqrt[3]{8x^4}}\) and \(\text{\sqrt[3]{27x^4}}\), we first recognize that the numbers 8 and 27 are perfect cubes (\(\text{2^3 = 8}\) and \(\text{3^3 = 27}\)). This means we can simplify these cube roots by breaking them into two parts:
  • 8 can be written as 2^3
  • 27 can be written as 3^3

By rewriting these numbers, we can simplify the cube roots as we did in the examples above.
Simplifying Expressions with Cube Roots
Simplifying expressions involves breaking them down into easier-to-manage parts. This becomes much easier when you identify factors of numbers that are perfect cubes.
Take the expression \(\text{\sqrt[3]{8x^4}}\).
Since 8 is \(\text{2^3}\), the cube root becomes \(\text{\sqrt[3]{2^3 x^4}}\), which simplifies to\(2 \sqrt[3]{x^4}\).
Similarly, the expression \(\text{\sqrt[3]{27x^4}}\), since 27 is \(\text{3^3}\), it simplifies to \(\text{3 \sqrt[3]{x^4}}\).
To combine like terms, we add \(\text{2 \sqrt[3]{x^4}}\) and \(\text{3 \sqrt[3]{x^4}}\) to get \(\text{5 \sqrt[3]{x^4}}\).
Role of Algebra in Simplifying Cube Roots
Algebra helps us manipulate symbols and solve equations systematically. When simplifying cube roots, algebraic rules guide us through breaking down and combining expressions.
For instance, with the cube roots in our example, algebra allows us to:
  • Recognize patterns and convert cubes back to their root terms
  • Combine like terms based on algebraic operators

By leveraging algebraic principles, we simplified \(\text{\sqrt[3]{8x^4}} + \sqrt[3]{27x^4}}\) to \(\text{5 \sqrt[3]{x^4}}\). This integration of algebraic concepts is crucial for solving complex expressions efficiently.

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Most popular questions from this chapter

Simplify each expression. $$\sqrt[3]{3 x^{5} y^{7}}-\sqrt[3]{24 x^{5} y^{7}}$$

Simplify. $$\frac{3}{\sqrt{2}-1}+\frac{4}{\sqrt{2}+1}$$

Solve each equation. $$(4 y-5)^{7}=0$$

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Discussion Which of the following equations are identities? Explain your answers. a) \(\sqrt{9 x}=3 \sqrt{x}\) b) \(\sqrt{9+x}=3+\sqrt{x}\) c) \(\sqrt{x-4}=\sqrt{x}-2\) d) \(\sqrt{\frac{x}{4}}=\frac{\sqrt{x}}{2}\)
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