To determine which two numbers [tex]$\sqrt{61}$[/tex] lies between on the number line, we need to approximate its value.
Step 1: Understand square roots.
- The square root of a number [tex]\(n\)[/tex] is a value that, when multiplied by itself, equals [tex]\(n\)[/tex].
- For example, [tex]\(\sqrt{49} = 7\)[/tex] because [tex]\(7 \times 7 = 49\)[/tex].
Step 2: Narrow down the range.
- Calculate some nearby squares to estimate the value of [tex]\(\sqrt{61}\)[/tex]:
- We know that [tex]\(7^2 = 49\)[/tex] and [tex]\(8^2=64\)[/tex].
- Since [tex]\(61\)[/tex] is between [tex]\(49\)[/tex] and [tex]\(64\)[/tex], [tex]\(\sqrt{61}\)[/tex] must be between [tex]\(7\)[/tex] and [tex]\(8\)[/tex].
Step 3: Further refine the range.
- Identify decimal values between 7 and 8:
- [tex]\(7.7^2 ≈ 59.29\)[/tex]
- [tex]\(7.8^2 ≈ 60.84\)[/tex]
- [tex]\(7.9^2 ≈ 62.41\)[/tex]
- Since [tex]\(61\)[/tex] is between [tex]\(60.84\)[/tex] and [tex]\(62.41\)[/tex], [tex]\(\sqrt{61}\)[/tex] is between [tex]\(7.8\)[/tex] and [tex]\(7.9\)[/tex].
Conclusion:
Therefore, the square root of 61 ([tex]\(\sqrt{61}\)[/tex]) lies between the numbers [tex]\(7.8\)[/tex] and [tex]\(7.9\)[/tex] on the number line.
