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Chapter 10: Problem 75

Approximate. Round to the nearest thousandth. $$ \sqrt[3]{(-3)^{5}} $$

Short Answer

Expert verified
-6.240

Step by step solution

01

Evaluate the exponent

First, calculate \[ (-3)^5 = -3 \times -3 \times -3 \times -3 \times -3 = -243 \] Therefore, \[ (-3)^5 = -243 \]
02

Find the cube root

Next, determine the cube root of -243. \ \[ \sqrt[3]{-243} \approx -6.240486 \]
03

Round to the nearest thousandth

Finally, round the result to the nearest thousandth. \ \[ -6.2405 \rightarrow -6.240 \]

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Key Concepts

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

Exponentiation
Exponentiation is a fundamental concept in algebra. It involves raising a base number to a certain power, represented as \text{} \(a^b\). Here, \(a\) is the base, and \(b\) is the exponent. In the exercise, the expression \((-3)^5\) means you're raising -3 to the 5th power. This means you multiply -3 by itself five times.
For example:
  • First, -3 times -3 equals 9.
  • Then, 9 times -3 equals -27.
  • Next, -27 times -3 equals 81.
  • Finally, -81 times -3 equals -243.
So, \((-3)^5 = -243\). Exponentiation is an efficient way to handle large repeated multiplications.
Cube Root
The cube root is the inverse operation of raising a number to the power of three. It answers the question: what number, when multiplied by itself three times, gives the original number?
Mathematically, it鈥檚 represented as \(\text{鈭泋}\). In the exercise, we need to find the cube root of -243, denoted as \( \text{鈭�(-243)} \).
Let's break it down:
  • Your goal is to find a number that, when cubed, equals -243.
  • Using a calculator or a detailed method, you find that it approximately equals -6.240486.
The cube root helps simplify some algebra problems and find 'hidden' values that correspond to an initial number.
Rounding to the nearest thousandth
Rounding numbers makes them easier to work with, especially when presenting data. The thousandth place is three digits to the right of the decimal point. Let's round -6.240486 to the nearest thousandth:
  • First, identify the thousandth place digit, which in this case is 0
  • Then look at the next digit to the right (the ten-thousandth place) which is 4.
  • If this digit is 5 or higher, you increase the thousandth place digit by 1. If 4 or lower, the thousandth place remains the same.Since 4 is less than 5, the thousandth digit remains 0
So, -6.240486 rounded to the nearest thousandth becomes -6.240. Rounding helps simplify answers while retaining a good level of accuracy.

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