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Chapter 10: Problem 20

Find all square roots of each number. \(\frac{81}{400}\)

Short Answer

Expert verified
The square roots are \(\frac{9}{20}\) and \(-\frac{9}{20}\).

Step by step solution

01

Understand the Problem

We need to find the square roots of the fraction \(\frac{81}{400}\). Remember that a square root of a number \(x\) is a number \(y\) such that \(y^2 = x\).
02

Write Down the Formula for Square Root of a Fraction

The square root of a fraction \(\frac{a}{b}\) is \(\frac{\text{sqrt}(a)}{\text{sqrt}(b)}\).
03

Calculate the Square Root of the Numerator

The numerator is 81. The square root of 81 is 9 because \(9^2 = 81\).
04

Calculate the Square Root of the Denominator

The denominator is 400. The square root of 400 is 20 because \(20^2 = 400\).
05

Combine the Results

Using the formula for the square root of a fraction, we find that the square roots of \(\frac{81}{400}\) are \(\frac{9}{20}\) and \(-\frac{9}{20}\). This is because both positive and negative values can be square roots.

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Key Concepts

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

Fractional Square Roots
Understanding fractional square roots can be very helpful. When you see a fraction like \(\frac{81}{400}\), you might wonder how to find its square root. Start by remembering that the square root of a fraction \(\frac{a}{b}\) is \(\frac{\text{sqrt}(a)}{\text{sqrt}(b)}\). This means you take the square root of the numerator and the square root of the denominator separately.

If you break it down:
  • Numerator: Take the square root of 81, which is 9.
  • Denominator: Take the square root of 400, which is 20.

Then, just put those results together as a fraction: \(\frac{9}{20}\). See how much easier it gets once you understand the process?
Numerator and Denominator
The terms numerator and denominator come up often in fractions. The numerator is the number above the fraction bar, and the denominator is the number below it.

For the fraction \(\frac{81}{400}\):
  • 81 is the numerator.
  • 400 is the denominator.

When finding the square root of a fraction, you need to find the square roots of both these parts separately. So, understanding each part will make the whole process smoother.
Positive and Negative Roots
Square roots come in pairs: a positive root and a negative root. This means that the square root of any positive number is always positive and negative.

For instance, if we have \(\frac{81}{400}\), you get two answers:
  • Positive root: \(\frac{9}{20}\)
  • Negative root: \(-\frac{9}{20}\)

This is because \((\frac{9}{20})^2 = \frac{81}{400}\) and \((-\frac{9}{20})^2 = \frac{81}{400}\). Therefore, both \(\frac{9}{20}\) and \(-\frac{9}{20}\) are valid square roots. Remembering this fact can help you find all possible solutions.

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