# Square Root of 16 — Value, Methods, Why It Is 4 TL;DR The square root of 16 is exactly 4 — rational, an integer, no decimal tail. The principal square root is 4; both 4 and -4 satisfy x² = 16. This article covers why √16 = 4, three methods (prime factorization, repeated subtraction, long division), where √16 shows up, and the slips students make on perfect-square radicals. 16 = √16 = 4 — exact, rational, integer. 16 is a perfect square because 4⋅4 = 16, so the answer comes out clean. **Result:** 16 = √16 = 4 (principal square root); both 4 and -4 satisfy x² = 16. **Notation:** Integer; radical form √16 simplifies to 4. **Method shown:** Prime factorization (Quick), repeated subtraction (Standard, with a Wrong-Path-First detour on the four-prime-factors question), long division (Stretch). **Approximate value:** No approximation needed — the answer is exact. **Exact form:** 4. ## Quick Reference Table — Small Perfect Squares | n | √n (exact) | Perfect square? | Rational / irrational | |----|---------------------|------------------|-----------------------| | 9 | 3 | Yes | Rational | | 12 | 3.4641 | No | Irrational | | 15 | 3.8730 | No | Irrational | | **16**| **4** | **Yes** | **Rational** | | 18 | 4.2426 | No | Irrational | | 20 | 4.4721 | No | Irrational | | 24 | 4.8990 | No | Irrational | | 25 | 5 | Yes | Rational | | 36 | 6 | Yes | Rational | √16 sits between √9 = 3 and √25 = 5 — the fourth-smallest non-trivial perfect square after 1, 4, 9. ## Where √16 appears √16 = 4 is the side length of any square with area 16 — a 4×4 grid, a flower bed laid out in a 4×4 pattern, a chessboard quadrant. The number 16 itself appears all over computing because 2⁴ = 16 — a hexadecimal digit takes values 0–15, exactly 16 possibilities, and 16-bit integers run from -32,768 to 32,767. In music, a 4/4 time signature carries 16 sixteenth-notes per measure, and √16 = 4 is the count of quarter-note beats that equal one measure. ## What "square root of 16" means The square root of a non-negative number n is the value x such that x² = n. For √16, the positive x with x² = 16 — which is 4, because 4⋅4 = 16. The √ symbol denotes the principal (non-negative) square root. The equation x² = 16 has two solutions, x = 4 and x = -4; the expression √16 refers only to the positive one. Same arithmetic, different question. ## Is √16 rational or irrational? √16 = 4 is **rational**. A number is a perfect square if and only if every prime in its factorization appears to an even power. 16 = 2⁴ — the 2 has exponent 4, which is even, so 16 is a perfect square and its square root is the integer 4. Every integer is rational (4 = 4/1), so √16 is rational. The decimal expansion of √16 is 4.000… — terminating immediately at the integer. ## How to find √16 — three methods ### Method 1 — Prime Factorization (Quick) Factor 16 into primes, then pair them. 16 = 2⋅2⋅2⋅2 = 2⁴ There are four 2s, forming two pairs. Each pair leaves the radical as a single 2: √16 = √(2⁴) = √(2² ⋅ 2²) = 2 ⋅ 2 = 4. **Final answer:** √16 = 4. ### Method 2 — Repeated Subtraction (Standard) A common student instinct: since 16 = 2⋅2⋅2⋅2 has four prime factors, the answer must somehow involve 4. Now run repeated subtraction. Subtract consecutive odd numbers from 16: 16−1=15, 15−3=12, 12−5=7, 7−7=0. Four subtractions, remainder zero — so √16 = 4. The method counts how many odd numbers (starting from 1) sum to 16, and that count is the square root for any perfect square. This is the correct reason: 1+3+5+7=16, four terms, so √16 = 4. **Final answer:** √16 = 4. ### Method 3 — Long Division (Stretch) Long division on a perfect square terminates immediately. Run it on 16.00. **Step 1.** Largest integer with square ≤ 16 is 4 (4² = 16). Subtract: 16−16=0. Bring down 000: 000. **Step 2.** Double 4: 8. Find d with (80+d)⋅d≤0. d=0. Subtract: 0−0=0. Every further step gives d = 0, so √16 = 4.000…=4. **Final answer:** √16 = 4. ## Three Errors That Cost The Most Marks on √16 ### 1. Confusing the square root expression with the quadratic equation **Where it slips in:** A student sees √16 and writes ±4, transferring the rule from x² = 16 ⇒ x = ±4. **Don't do this:** 16=±4 — written as the final answer to a square-root expression. **The correct way:** √16 = 4 (the principal, non-negative root). The ± shows up only when solving x²=16, because taking the square root of both sides introduces ± explicitly: x=±√16. ### 2. Reading √16 as division **Where it slips in:** A student rushes and answers 8 (16÷2) or 4 (16÷4), right answer wrong method). **Don't do this:** √16 is the number whose square equals 16. ### 3. Writing √16 when 4 is expected **Where it slips in:** A student leaves the answer as √16 in a multiple-step problem, then later squares it expecting 16 — but marks are deducted. **The correct way:** √16 = 4. For a perfect square, always resolve to the integer. The radical form √16 is only acceptable as an intermediate expression; the final answer must show 4. ## The Short Version - The **square root of 16** is 4 — the principal, non-negative root. - √16 is rational because 16 is a perfect square (16=2⁴, every prime to an even power). - The expression √16 has a single value (4); the equation x²=16 has two solutions (±4). - All three methods reach the same answer — prime factorization in one line, repeated subtraction in four steps, long division terminating immediately. - Always square back: 4⋅4=16 ✓. If the squaring-back check fails, the square-root answer is wrong. ## Frequently Asked Questions What is the square root of 16? 4 Is the square root of 16 rational? Yes. √16 = 4 is an integer, and every integer is rational. Why is √16 equal to 4 and not ±4? The √ symbol denotes the principal (non-negative) square root by convention. The equation x²=16 has two solutions, ±4, but √16 refers only to +4. Is 16 a perfect square? Yes. 16=4², every prime factor appears to an even power. What is the square root of −16? Not a real number — √−16 = 4i, where i is the imaginary unit. Real-number square roots are defined only for non-negative inputs.