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Chapter 1: Problem 16

Simplify the given expressions. In each of \(5-9\) and \(12-21,\) the result is an integer. $$-\sqrt[5]{-32}$$

Short Answer

Expert verified
The simplified expression is 2.

Step by step solution

01

Determine the Root

The expression given is \( -\sqrt[5]{-32} \). We are tasked with finding the fifth root of \(-32\). Since this is an odd root, the negative sign within the root indicates we are looking for a negative number that, when raised to the power of 5, equals \(-32\).
02

Find the Fifth Root of -32

We need to determine what number raised to the fifth power results in \(-32\). Calculate: \((-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\). Thus, \(-32\) can be expressed as \((-2)^5\).
03

Apply the Root to Simplify the Expression

Apply the fifth root to the expression \( - \sqrt[5]{-32} \). Given our calculation, we know \( \sqrt[5]{-32} = -2 \). Thus, the original expression becomes: \( -(-2) \).
04

Simplify the Expression

Simplify the expression \(-(-2)\) which results in positive \(2\).

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Key Concepts

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

Integer Outcomes
Simplifying expressions often results in integer outcomes, especially when the root of the number is a perfect power of that root degree. An integer outcome is simply a whole number without any fractions or decimals. Let's understand easily:
  • Whole numbers in the sense of integers include positives, negatives, and zero.
  • We say an outcome is an integer if it's not fractional or decimal.
  • For expressions involving roots, if the number is a perfect power, the root will yield an integer.
In our exercise, the expression simplifies to the integer 2 because the fifth root of -32 is -2, and when applying the external negative sign, it results in a positive integer.
Fifth Roots
Calculating roots can sometimes be challenging, but fifth roots can be easier if you know the right approach.
  • To find the fifth root of a number, you need to identify a number that, when raised to the power of five, gives the original number.
  • This task becomes quite straightforward if you recognize the original number as a perfect fifth power.
  • Odd roots like the fifth root allow us to handle negative numbers neatly. For example, the fifth root of -32 is -2.
By noting that \((-2)^5 = -32\), we see how roots of negative numbers can yield clean results when dealing with odd powers.
Negative Numbers
Negative numbers play a special role when dealing with roots and exponents, adding an extra layer to simplifying expressions.
  • Negative numbers have unique properties. For instance, multiplying an odd number of negative signs results in a negative outcome, which is useful for root calculations.
  • When calculating roots, odd roots of negative numbers result in negative integers.
  • Always pay attention to negative signs; they can change the math dramatically, as seen in our example where the negative fifth root of -32 simplifies the expression neatly to a positive integer due to the double negation.
So, using negative numbers appropriately helps simplify expressions, like turning negative fifth roots into simple, positive outcomes.

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