AI Tutor
If someone asks you for 25\sqrt{25}, you instantly say 55. But what if they ask you for 29\sqrt{29}?
Because 2929 is not a perfect square (an integer that results from squaring another integer), its square root is irrational — an unending, non-repeating decimal.
How can we pinpoint where 29\sqrt{29} lives on the number line without touching a calculator?

Bounding Between Perfect Squares

Every irrational square root is trapped between two consecutive integer square roots. To estimate N\sqrt{N}, we first find the closest perfect square below NN and the closest perfect square above NN.
Since 25<29<3625 < 29 < 36, taking square roots across the inequality gives 25<29<36\sqrt{25} < \sqrt{29} < \sqrt{36}, which simplifies to 5<29<65 < \sqrt{29} < 6.
Knowing that 29\sqrt{29} is between 55 and 66 is a good start, but how do we estimate the exact tenths digit?

Estimating to the Nearest Tenth

We use linear interpolation, which treats the gap between perfect squares as a proportional fraction: Estimate≈a+N−a2b2−a2\text{Estimate} \approx a + \frac{N - a^2}{b^2 - a^2}.
To verify whether 5.45.4 is accurate to the nearest tenth, we test neighboring tenths by squaring: 5.32=28.095.3^2 = 28.09 and 5.42=29.165.4^2 = 29.16.
Since 29.1629.16 is much closer to 2929 than 28.0928.09, 29≈5.4\sqrt{29} \approx 5.4 to the nearest tenth.