\(0.81\) Square Root
The expression \(0.81\) square root means the principal square root of \(0.81\), written as \(\sqrt{0.81}\). The principal square root is the nonnegative number whose square equals \(0.81\).
Result: \(\sqrt{0.81}=0.9\).
1) Rewrite \(0.81\) as a Fraction
Since \(0.81\) has two decimal places, it can be written as a fraction over 100: \(0.81=\frac{81}{100}\).
2) Apply the Square Root to Numerator and Denominator
Use the rule \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) for \(a\ge 0\) and \(b>0\). Then \(\sqrt{0.81}=\sqrt{\frac{81}{100}}=\frac{\sqrt{81}}{\sqrt{100}}\).
3) Evaluate the Perfect Squares
The numbers 81 and 100 are perfect squares: \(\sqrt{81}=9\) and \(\sqrt{100}=10\). Therefore \(\sqrt{0.81}=\frac{9}{10}=0.9\).
\[ \sqrt{0.81} =\sqrt{\frac{81}{100}} =\frac{\sqrt{81}}{\sqrt{100}} =\frac{9}{10} =0.9 \]
4) Quick Check by Squaring
A reliable algebra check is to square the answer: \(0.9^2=0.81\). Since the square matches the original number, \(\sqrt{0.81}=0.9\) is correct.
5) Visualization: Square-Root Check on a Number Line
The value of \(\sqrt{0.81}\) is \(0.9\), since \(0.81=\frac{81}{100}\) and \(\sqrt{\frac{81}{100}}=\frac{9}{10}=0.9\).