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Chapter 10: Problem 19

Use radical notation to write each expression. Simplify if possible. $$ (-64)^{2 / 3} $$

Short Answer

Expert verified
The simplified expression is 16.

Step by step solution

01

Understand the Expression

The given expression is \((-64)^{2/3}\). The fraction \(\frac{2}{3}\) as an exponent means the cube root squared.
02

Rewrite Using Radical Notation

Rewriting the expression in terms of radicals, we have: \(((-64)^{1/3})^2= (\sqrt[3]{-64})^2\).This represents the cube root of \(-64\) squared.
03

Evaluate the Cube Root

Find the cube root of \(-64\). You know that \(-4\times -4 \times -4 = -64\). Therefore, \(\sqrt[3]{-64} = -4\).
04

Square the Result

Square the result from the cube root calculation. That is: \((-4)^2 = 16\).
05

Solution Verification

Verify the simplification by recalculating if necessary. Re-evaluation confirms that the steps lead to the consistent answer of \(16\).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation that simplifies the process of repeated multiplication of a number. It's expressed with a number, known as the base, and a smaller number written above and to the right, called the exponent.

For example, in the expression \(-64^{2/3}\), \(-64\) is the base and \( rac{2}{3}\) is the exponent. The exponent tells you how many times to multiply the base by itself. In this case:
  • The numerator (2) means you'll square the result of the base raised to the power of the denominator (3).
  • The denominator (3) indicates you're looking to take the cube root of the base.
By understanding the function of the exponent, we can break down complex expressions into simpler, smaller parts for easier calculation.
Rational Exponents
Rational exponents involve fractions as exponents, which can be seen as a way to express both roots and powers in one expression. The expression \(-64^{2/3}\) is a perfect example.
  • The rational exponent \( rac{2}{3}\) suggests that we first take the cube root of \(-64\), then square the result.
  • Rational exponents are useful because they allow one to rewrite expressions involving roots in a uniform manner.
This notation not only simplifies calculations but can also make solving algebraic expressions easier by allowing for the combination of different operations in one step.
Cube Root
The cube root of a number is one of its roots that, when multiplied by itself twice, results in the original number. In mathematical terms, if \(x^3 = a\), then \(x\) is said to be a cube root of \(a\).
  • For \(-64\), the cube root is \(-4\), since \(-4 \times -4 \times -4 = -64\).
  • Cube roots are special because they can apply to both positive and negative numbers, while square roots are generally only for non-negative numbers without resulting in imaginary numbers.
Understanding cube roots is crucial in simplifying expressions that contain rational exponents with \(3\) in the denominator, just as in the original exercise.
Simplification
Simplification in mathematics means expressing a mathematical idea in its simplest form. It involves combining like terms, reducing fractions, and executing any arithmetic operations involved.
  • For the expression \((-64)^{2/3}\), simplification occurs by first finding the cube root \(-4\) and then squaring it to reach \(16\).
  • This step-by-step approach reduces the expression to its simplest form, making it easier to understand and work with.
Ultimately, simplification makes problem-solving more efficient and helps convey solutions in a clear, concise manner.

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Most popular questions from this chapter

Use a calculator to write a four-decimal-place approximation of each number. $$ 20^{1 / 5} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[4]{(x+3)^{2}} $$

Use a calculator to write a four-decimal-place approximation of each number. $$ 8^{1 / 4} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y^{2}} \cdot \sqrt[6]{y} $$

Which of the following are not real numbers? $$\sqrt{-17}$$
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