Problem 36 Write as a radical and then simp... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 8: Problem 36

Write as a radical and then simplify. $(-64) 1 / 3$

Short Answer

Expert verified

-4

Step by step solution

Convert to Radical Form

First, express the expression $(-64)^{1/3}$ as a radical. A fractional exponent with 1/3 is equivalent to the cube root. Thus, $(-64)^{1/3}$ becomes $\sqrt[3]{-64}$.

Simplify the Radical Expression

Identify the cube root of $-64$. Since $-64 = (-4) \times (-4) \times (-4)$, the cube root of $-64$ is $-4$. Therefore, $\sqrt[3]{-64} = -4$.

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Key Concepts

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

Fractional Exponents

Fractional exponents represent another way to express roots, such as square roots or cube roots. They offer a neat, compact symbolic form. Instead of $\sqrt[3]{-64}$, we use $(-64)^{1/3}$. Here, the exponent \frac{1}{3}\ suggests taking the cube root.

Understanding fractional exponents is essential for simplifying radical expressions and performing various mathematical calculations efficiently. It is crucial to remember that the numerator in the fractional exponent signifies the power to which the base is raised, whereas the denominator indicates the root to be taken of that base.

Cube Roots

A cube root asks the question, "what number multiplied by itself twice gives us the original number?" In our exercise, $-64$, we look for a number that when multiplied by itself twice results in $-64$. The cube root is denoted by \sqrt[3]{}\.

In $(-64)^{1/3} = \sqrt[3]{-64}$, we had to find the number that multiplied three times results in $-64$. Knowing that $-4)$ multiplied by itself twice gives us $-64$, we can confidently say \sqrt[3]{-64} = -4\. Cube roots are fundamental when dealing with negative numbers, as they maintain consistent results, unlike even roots which may not accept negative radicands.

Simplifying Radicals

Simplifying radicals involves reducing them to their most simplified form. This often means recognizing perfect squares or perfect cubes within the radicand, like in $-64$. Breaking $-64$ down to factors of $(-4)$ indicates it is a perfect cube.

By identifying this, the radical expression \sqrt[3]{-64}\ can quickly simplify to $-4$. Simplifying not only makes expressions easier to work with but also allows for easier further computations or algebraic manipulation. Familiarity with factorization and identifying cubes or higher roots simplification will empower your mathematics toolkit.

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91影视

Write as a radical and then simplify. \((-64) 1 / 3\)

Short Answer

Step by step solution

Convert to Radical Form

Simplify the Radical Expression

Key Concepts

Fractional Exponents

Cube Roots

Simplifying Radicals

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