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Chapter 8: Problem 55

Write as a radical and then simplify. \((-32) 3 / 5\)

Short Answer

Expert verified
The simplified form is \(-8\).

Step by step solution

01

Express as a Radical

Begin by expressing the expression \( (-32)^{\frac{3}{5}} \) as a radical. The fractional exponent \( \frac{3}{5} \) indicates a combination of taking the fifth root and raising to the third power. Thus, it can be rewritten as:\[ \left( \sqrt[5]{-32} \right)^3 \]
02

Simplify the Radical

Find the fifth root of \(-32\). Since \( (-2)^5 = -32 \),the fifth root of \(-32\) is \(-2\). This simplifies to:\[ \left( -2 \right)^3 \]
03

Calculate the Power

Raise \(-2\) to the third power:\[ (-2)^3 = -8 \] Thus, the simplified form of the expression is \(-8\).

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Key Concepts

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

Fractional Exponents
Fractional exponents are an exciting way to express roots and powers simultaneously. In mathematics, a fractional exponent like \( a^{\frac{m}{n}} \) signifies two operations occurring together.
  • The numerator \( m \) indicates that the base \( a \) should be raised to the power of \( m \).
  • The denominator \( n \) indicates taking the nth root of the base \( a \).

This expression can be written as the root raised to a power: \( \left(\sqrt[n]{a}\right)^m \).
In our provided problem, \( (-32)^{\frac{3}{5}} \), the exponent \( \frac{3}{5} \) tells us to:
  • Take the fifth root of -32.
  • Then raise the result to the third power.

Understanding fractional exponents is essential as they offer a compact way to solve complex expressions.
Fifth Roots
Finding the fifth root of a number means determining which number, when multiplied by itself five times, returns the original number. When we look at our example of \( -32 \), we seek a number whose fifth power results in \( -32 \).
  • The solution is \( -2 \), since: \((-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\).

Remember that the fifth root of a negative number remains negative because multiplying five negative values results in a negative product.
Fifth roots can be visualized for various calculations, and recognizing roots is crucial for simplifying radicals.
Simplifying Radicals
Simplifying radicals involves reducing expressions to their simplest form while still showcasing the root's mathematical uniqueness. A radical often appears in expressions with fractional exponents, and simplifying them aids in ease of calculation.
The simplification process requires precision to ensure correctness:
  • Identify and compute the root of the number.
  • Raise the result to the given power if necessary.

Let's continue our process from our exercise.
  • We found that \( \sqrt[5]{-32} = -2 \).
  • Then, we raised this result to the third power: \( (-2)^3 = -8 \).

By leading these steps consistently, we have effectively simplified the radical expression \( (-32)^{\frac{3}{5}} \) to \( -8 \). This boosts not only your understanding but lets radicals be an integrated aspect of mathematical problem-solving.

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Most popular questions from this chapter

Use the distance formula to calculate the distance between the given two points. $$ (-3,-4) \text { and }(3,-6) $$

Simplify. (Assume all radicands containing variable expressions are positive.) \(125 x y 3------\sqrt{3}+8 x 3 y---\sqrt{3}-216 x y 3------\sqrt{3}+10 x \cdot y \sqrt{3}\)

Research and discuss some of the reasons why it is a common practice to rationalize the denominator.

Simplify. \(135---\sqrt{3}+40--\sqrt{3}-5 \sqrt{3}\)

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