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Chapter 7: Problem 92

Simplify. $$\sqrt[3]{16 a^{4}}+a \sqrt[3]{2 a}$$

Short Answer

Expert verified
The simplified expression is \(3a\sqrt[3]{2a}\).

Step by step solution

01

Factor Inside the Radicals

Rewrite each term to make the simplification process easier.For \(\sqrt[3]{16a^4}\), note that 16 can be factored into \(2^4\).Therefore, rewrite \(\sqrt[3]{16a^4}\) as \(\sqrt[3]{2^4a^4}\). For \(a\sqrt[3]{2a}\), there is no change required.
02

Apply the Cube Root to Each Factor Separately

Apply the cube root to each part of the expression inside the radicals.\(\sqrt[3]{2^4a^4} = \sqrt[3]{2^3 \cdot 2a^3 \cdot a} = 2a \cdot \sqrt[3]{2a} \).Now the expression becomes \(2a\sqrt[3]{2a} + a\sqrt[3]{2a}\).
03

Combine Like Terms

Notice that both terms have \(\sqrt[3]{2a}\) as a factor. Combine the coefficients of like terms by addition: \(2a\sqrt[3]{2a} + a\sqrt[3]{2a} = (2a + a)\sqrt[3]{2a} = 3a\sqrt[3]{2a}\).

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Key Concepts

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

cube root
When dealing with cube roots, we are looking for a number that, when multiplied by itself three times, gives us the original number. Mathematically, the cube root of a number N is represented as \(\sqrt[3]{N}\). For example, the cube root of 8 is 2, because \(2 \cdot 2 \cdot 2 = 8\).
factoring
Factoring involves breaking down a number or expression into its prime components. For instance, the number 16 can be expressed as \(2^4\) because \(2 \cdot 2 \cdot 2 \cdot 2 = 16\). Likewise, the expression \(16a^4\) can be rewritten as \(2^4a^4\). This helps in simplifying radicals by applying cube roots individually to each factor.
like terms
Like terms are terms that have identical variables raised to the same power. In the context of the equation \(2a\sqrt[3]{2a} + a\sqrt[3]{2a}\), both terms share the factor \(\sqrt[3]{2a}\). To combine these, you simply add the coefficients of the like terms: 2a and a. This leads to \(3a\sqrt[3]{2a}\).

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Most popular questions from this chapter

Discussion Which one of the following expressions is not equivalent to the others? a) \((\sqrt[3]{x})^{4}\) b) \(\sqrt[4]{x^{3}}\) c) \(\sqrt[3]{x^{4}}\) d) \(x^{4 / 3}\) e) \(\left(x^{1 / 3}\right)^{4}\)

Work in a small group to write a formula that gives the side of a square in terms of the diagonal of the square and explain the formula to the other groups.

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. Find the length of the side of a square sign whose area is 50 square feet.

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. The sail area-displacement ratio \(S\) provides a measure of the sail power available to drive a boat. For a boat with a displacement of \(d\) pounds and a sail area of \(A\) square feet \(S\) is determined by the function $$S=16 A d^{-2 / 3}$$ a) Find \(S\) to the nearest tenth for the Tartan \(4100,\) which has a sail area of 810 square feet and a displacement of \(23,245\) pounds. b) Write \(d\) as a function of \(A\) and \(S\).

Solve each equation. $$\sqrt[3]{2-w}=\sqrt[3]{2 w-28}$$
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