Question 5 (1 point)
Saved
Between which two consecutive integers on a number line would you locate $\sqrt[3]{-18}$ ?
11
12
-3 and -4
-2 and -3
-1 and -2
2 and 3
18
Question 6 (1 point)
$\checkmark$ Saved
Which of these numbers is irrational?
$\sqrt{\frac{49}{16}}$
$\sqrt{48}$
$-68$
$\sqrt[3]{216}$
Answer (1)
Determine the consecutive integers for 3 − 18 : Since ( − 3 ) 3 = − 27 < − 18 < − 8 = ( − 2 ) 3 , 3 − 18 lies between -3 and -2.
Identify the irrational number: Check each option to see if it can be expressed as a fraction of two integers.
16 49 = 4 7 (rational).
48 = 4 3 (irrational).
− 68 (rational).
3 216 = 6 (rational).
The consecutive integers are − 3 and − 2 and the irrational number is 48 .
Explanation
Problem Analysis We are asked to locate 3 − 18 between two consecutive integers and to identify an irrational number from a given list.
Locating the Cube Root First, let's find the two consecutive integers between which 3 − 18 lies. Since 3 − 18 is negative, we are looking for two consecutive negative integers. We can estimate the value of 3 − 18 . We know that ( − 2 ) 3 = − 8 and ( − 3 ) 3 = − 27 . Since − 27 < − 18 < − 8 , we have − 3 < 3 − 18 < − 2 . To confirm, we can calculate the cube root of -18, which is approximately -2.62. Therefore, 3 − 18 lies between -3 and -2.
Identifying Irrational Numbers Now, let's identify the irrational number from the list: 16 49 , 48 , − 68 , 3 216 . Recall that an irrational number cannot be expressed as a fraction q p where p and q are integers and q = 0 .
Checking the First Number 16 49 = 4 7 , which is a rational number.
Checking the Second Number 48 = 16 ⋅ 3 = 4 3 . Since 3 is irrational, 4 3 is also irrational.
Checking the Third Number − 68 is an integer, so it is a rational number.
Checking the Fourth Number 3 216 = 6 , which is a rational number.
Final Answer Therefore, the irrational number is 48 .
Examples
Understanding irrational numbers and cube roots is crucial in various fields. For instance, in engineering, calculating the volume of a cube-shaped structure might involve cube roots. If the volume is not a perfect cube (like -18 in our problem), the side length will be an irrational number. Similarly, when designing structures, engineers need to work with irrational numbers to ensure precision and stability. Knowing how to locate these numbers on a number line helps in visualizing and approximating values for practical applications.
