Problem 518 In the following exercises, simp... [FREE SOLUTION]

Chapter 9: Problem 518

In the following exercises, simplify. \(\sqrt[3]{64 a^{10}}-\sqrt[3]{-216 a^{12}}\)

Short Answer

Expert verified

4 a^{\frac{10}{3}} + 6 a^4

Step by step solution

Simplify each cube root separately

First, simplify \ \ \( \sqrt[3]{64 a^{10}} \) \ 64 is a perfect cube because \ \( 64 = 4^3 \). Additionally, use the property of exponents: \( \sqrt[3]{a^{10}} = a^{\frac{10}{3}} \). Therefore, \ \( \sqrt[3]{64 a^{10}} = \sqrt[3]{4^3 a^{10}} = 4 a^{\frac{10}{3}} \).

Simplify the second cube root

Next, simplify \ \ \( \sqrt[3]{-216 a^{12}} \) \ -216 is a perfect cube because \ \( -216 = (-6)^3 \). Additionally, use the property of exponents: \( \sqrt[3]{a^{12}} = a^{\frac{12}{3}} \). Therefore, \ \( \sqrt[3]{-216 a^{12}} = \sqrt[3]{(-6)^3 a^{12}} = -6 a^4 \).

Subtract the simplified expressions

Finally, subtract the simplified cube roots: \ 4 a^{\frac{10}{3}} - (-6 a^4) \ Since \( a^{\frac{10}{3}} \) and \( a^4 \) can't be combined directly, the simplified form is: \ 4 a^{\frac{10}{3}} + 6 a^4.

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Key Concepts

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

Cube Roots

In math, the cube root of a number is a value that, when multiplied by itself three times, gives the original number. Think of it as the opposite of cubing a number.
If you see \( \sqrt[3]{x} \)\, it means 'what number, when cubed, equals x?'

For example, \( \sqrt[3]{64} = 4 \)\, because \( 4^3 = 64 \).
Similarly, \( \sqrt[3]{-216} = -6 \)\, because \( (-6)^3 = -216 \).

Property of Exponents

Exponents are a way to denote repeated multiplication of a number by itself. Understanding their properties is crucial.
One useful property is \( {a^{m \over n}} \)\, which means taking the mth power and nth root of a.

When working with cube roots and exponents, remember these key points:

The cube root of a number with an exponent can be simplified as \( \sqrt[3]{a^b} = a^{\frac{b}{3}} \)\.
You can see this in the example: \( \sqrt[3]{a^{10}} = a^{\frac{10}{3}} \).
Another example is \( \sqrt[3]{a^{12}} = a^{4} \).

Perfect Cubes

A perfect cube is a number that can be written as the cube of an integer.
For instance,
\( 64 = 4^3 \)\ , and \( -216 = (-6)^3 \)\ .

Let's list out a few perfect cubes to help you recognize them more easily:

1 = \( 1^3 \).
8 = \( 2^3 \).
27 = \( 3^3 \).
64 = \( 4^3 \).
125 = \( 5^3 \).
216 = \( 6^3 \).
-1, -8, -27, -64, -125, -216 are also perfect cubes of negative numbers.

By identifying perfect cubes, the task of simplifying expressions becomes easier.

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91影视

In the following exercises, simplify. \(\sqrt[3]{64 a^{10}}-\sqrt[3]{-216 a^{12}}\)

Short Answer

Step by step solution

Simplify each cube root separately

Simplify the second cube root

Subtract the simplified expressions

Key Concepts

Cube Roots

Property of Exponents

Perfect Cubes

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