Simplification

What you need about roots

A square root asks: which number times itself gives this? $\sqrt{11025} = 105$ because $105 \times 105 = 11025$. A cube root asks: which number cubed gives this? $\sqrt[3]{9261} = 21$.

Spotting and finding perfect squares

• A perfect square never ends in 2, 3, 7 or 8, and never in an odd number of zeros.
• Last-digit map (root → square): 1 or 9 → 1; 2 or 8 → 4; 3 or 7 → 9; 4 or 6 → 6; 5 → 5; 0 → 0.
• To find $\sqrt{11025}$: pair digits from the right (1 | 10 | 25). First pair: $1^2 = 1$ → first digit 1. Double it (2); find digit $d$ so that $2d \times d$ fits 10 → $d = 4$ ($24 \times 4 = 96$)? 96 fits under 110? wait — work with 010 remainder 10: bring down 25 → 1025; $24\Box \times \Box \le 1025$ → $245 \times 5 = 1225$ too big, $244 \times 4 = 976$ ✓ → root 105? The long-division method always lands exactly; in exams you mostly match options with the last-digit map and a rough size check.

Size check: count digit pairs. A 5-digit number (like 11025 → pairs 1, 10, 25) has a 3-digit root. Between $100^2 = 10000$ and $110^2 = 12100$; last digit 5 → root ends in 5 → 105.

Cube roots in seconds

Group digits in threes from the right. The number of groups = number of root digits.

• Unit digit of cube → unit digit of root: 1↔1, 4↔4, 5↔5, 6↔6, 9↔9, 0↔0, and swaps 2↔8, 3↔7. Example: $\sqrt[3]{9261}$: groups 9 | 261. Cube ends in 1 → root ends in 1. $2^3 = 8 \le 9 < 27 = 3^3$ → first digit 2. Root = 21.

Roots of decimals

The decimal point just shifts: $\sqrt{0.0169} = 0.13$ (2 decimal places in → 1 in the root); $\sqrt[3]{0.000064} = 0.04$ (6 places → 2 places).

Least number to make a perfect square

"Find the least number to be subtracted from N": take $\sqrt{N}$, keep the whole part $k$, answer $= N - k^2$. "Least number to be added": use $(k+1)^2$; answer $= (k+1)^2 - N$. Example: $N = 1300$: $36^2 = 1296 < 1300 < 1369 = 37^2$ → add 69.

Nested radicals

• Infinite $\sqrt{a+\sqrt{a+\dots}}$ equals the positive root of $x^2 = a + x$. For $a = 6$: $x^2 - x - 6 = 0$ → $x = 3$.
• Infinite $\sqrt{a \times \sqrt{a \times \dots}}$ equals $a$.
• Finite nests are worked from the inside out: $\sqrt{30 + \sqrt{24 + \sqrt{144}}} = \sqrt{30 + \sqrt{36}} = \sqrt{36} = 6$.
• Quick identity: if $n = k(k+1)$ then $\sqrt{n + \sqrt{n + \dots}} = k+1$ (e.g. 12 = 3 × 4 → 4) and $\sqrt{n - \sqrt{n - \dots}} = k$.

Root equations

Move the root to one side, square both sides once: $\sqrt{1 + \frac{x}{225}} = \frac{16}{15}$ → $1 + \frac{x}{225} = \frac{256}{225}$ → $x = 31$.

Quick revision

• Perfect squares: last digit in {0,1,4,5,6,9}; size from digit pairs; exact digit from the map.
• Cube root: groups of three digits; unit-digit swap map.
• Subtract → $N - k^2$; add → $(k+1)^2 - N$.
• Infinite $\sqrt{a+\sqrt{a+\dots}}$: solve $x^2 = a + x$.
• Nested roots: inside out.