Between which two numbers does √72 lie on a number line?
The Correct Answer and Explanation is :
To find between which two numbers ( \sqrt{72} ) lies on a number line, we begin by simplifying the square root.
Step 1: Simplify ( \sqrt{72} )
Start by factoring 72 into its prime factors:
[
72 = 36 \times 2
]
Since ( \sqrt{36} = 6 ), we can simplify the square root as follows:
[
\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6 \times \sqrt{2}
]
Now we need to estimate ( \sqrt{2} ). ( \sqrt{2} ) is approximately 1.414. Thus:
[
\sqrt{72} = 6 \times 1.414 \approx 8.484
]
Step 2: Identify the two numbers on the number line
To determine between which two whole numbers ( \sqrt{72} ) lies, we can use the fact that:
- ( 8^2 = 64 )
- ( 9^2 = 81 )
Since ( \sqrt{72} ) is between the squares of 8 and 9, we conclude that:
[
8 < \sqrt{72} < 9
]
Thus, ( \sqrt{72} ) lies between 8 and 9 on the number line.
Step 3: Understanding the reasoning
We started by simplifying ( \sqrt{72} ) and approximating the value of ( \sqrt{2} ), which gave us 8.484. This value is greater than 8 but less than 9. The square roots of perfect squares, such as ( \sqrt{64} = 8 ) and ( \sqrt{81} = 9 ), help us place ( \sqrt{72} ) between these two values. This process shows how approximations and properties of square roots can be used to estimate where a number lies on a number line.