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Chapter 3: Problem 28

In Exercises 28鈥�35, find the indicated roots without using a calculator. the cube root of \(-216\)

Short Answer

Expert verified
The cube root of \(-216\) is \-6\.

Step by step solution

01

- Understand the problem

The problem requires finding the cube root of \(-216\). A cube root asks what number, when multiplied by itself three times, results in \(-216\).
02

- Consider properties of cube roots

Recall that the cube root of a negative number is also negative. This is because a negative number multiplied by itself three times remains negative.
03

- Break down the number

Consider the absolute value \(216\) and find its prime factorization to help determine its cube root. \(216 = 6 \times 6 \times 6 = 6^3\)
04

- Apply the cube root

Since \(216 = 6^3\), the cube root of \216\ is \6\. Therefore, the cube root of \(-216\) is \-6\ because \(-6 \times -6 \times -6 = -216\).

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Key Concepts

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

Understanding Negative Numbers
Negative numbers are numbers less than zero and are represented with a minus sign (e.g., -1, -2, -3). They play a crucial role in mathematics, especially when working with roots and exponents. Here are some important points to remember about negative numbers:
  • When you multiply an odd number of negative numbers, the result is negative (e.g., ewlineewline -2 脳 -2 脳 -2 = -8).
  • When you multiply an even number of negative numbers, the result is positive (e.g., ewlineewline -2 脳 -2 = 4).
Because of these properties, the cube root of a negative number is always negative. Knowing this helps to solve problems like finding the cube root of ewlineewline-216 effectively.
Prime Factorization
Prime factorization involves breaking down a number into its prime number factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7). To find the prime factorization of a number, you repeatedly divide by the smallest prime number until you reach 1.
Here's how to find the prime factorization of 216:
  • 216 can be divided by 2: ewlineewline 216 梅 2 = 108.
  • 108 can be divided by 2: ewlineewline 108 梅 2 = 54.
  • 54 can be divided by 2: ewlineewline 54 梅 2 = 27.
  • 27 can be divided by 3: ewlineewline 27 梅 3 = 9.
  • 9 can be divided by 3: ewlineewline 9 梅 3 = 3.
  • Finally, 3 can be divided by 3:ewline ewline3 梅 3 = 1.
So, the prime factorization of 216 is: ewlineewline 2^3 脳 3^3.
Properties of Exponents
Exponents are used to represent repeated multiplication of a number by itself. In the expression ewlineewline 6^3, 6 is the base and 3 is the exponent, which means 6 multiplied by itself three times (6 脳 6 脳 6). Here are some fundamental properties of exponents to keep in mind:
  • Product of powers: When we multiply two powers with the same base, we add the exponents:ewlineewline a^m * a^n = a^(m+n).
  • Power of a power: When we take a power to another power, we multiply the exponents:ewlineewline (a^m)^n = a^(m*n).
  • Power of a product: When we take the power of a product, we apply the exponent to each factor inside the parentheses:ewlineewline (ab)^n = a^n * b^n.
  • Negative exponents: A negative exponent represents the reciprocal of the base raised to the positive exponent:ewlineewline a^(-n) = 1/a^n.
Understanding these properties helps in solving problems like finding the cube roots of numbers. For instance, if you know that 216 is 6^3, then taking the cube root of 216 simply means reversing the exponentiation, resulting in 6. Likewise, doing the same for -216 results in -6, highlighting the power of understanding exponent rules.

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Most popular questions from this chapter

Julia found three Web sites that sell 4-inch-square stickers of her favorite band鈥檚 logo. The three sites sell the stickers for different prices, and charge different amounts for shipping. Site \(1 :\) Stickers are 75 each; shipping is \(\$ 4\) for any size order. Site \(2 :\) Stickers are 60\(\phi\) each; shipping is \(\$ 5.50\) for any size order. Site \(3 :\) Stickers are \(\$ 1.25 ;\) shipping is included. a. For each site, write an equation to represent the charge \(C\) for ordering any number of stickers \(s\) b. Graph your three equations on axes like these. Label each graph with its site number. c. Use your graph to answer this question: If Julia wants to order 16 stickers, which site will charge her the least? d. Use your graph to answer this question: If Julia wants to order 10 stickers, which site will charge her the least?

Solve each equation. If it has no solution, write "no solution." $$ \sqrt{x-3}=9 $$

Rewrite each expression as simply as you can. $$m^{-3} \cdot m^{4} \cdot b^{7}$$

Find the indicated roots without using a calculator. $$ \sqrt{0.0064} $$

Determine whether the three points in each set are collinear. $$ (1,-1.6),(5,0),(6,0.4) $$
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