How to Convert Radicals to Decimals Without a Calculator

To convert a radical to a decimal without a calculator, you usually estimate its value and refine the estimate. This works because squaring a positive decimal lets you check whether it is below or above the desired square root.

The methods below focus on square roots, which are the most common radicals in this context. They work with written arithmetic, though longer decimals require more calculation. A root may be exact, such as $\sqrt{49}=7$, or it may have a nonterminating decimal expansion, such as $\sqrt{2}$; in the latter case, report an approximation to the number of places you need.

Simplify the radical first

Look for perfect-square factors inside the radical. For example,

$$\sqrt{72}=\sqrt{36\cdot 2}=6\sqrt{2}.$$

This reduces the size of the calculation. It also helps you spot exact answers: if the radicand is a perfect-square integer, its root is an integer. More generally, simplifying a radical may reveal an exact rational value, as in $\sqrt{1/4}=1/2$.

Estimate by bracketing

For a positive number $N$, if $N$ is a perfect-square integer, its root is an integer. Otherwise, find consecutive integers whose squares lie on either side of $N$. If $a^2<N<b^2$, then $a<\sqrt{N}<b$. Test decimal numbers between $a$ and $b$ by squaring them. Each test narrows the interval.

Example: estimate $\sqrt{7}$

Since $2^2=4$ and $3^2=9$, the answer lies between 2 and 3. Test hundredths:

$$2.64^2=6.9696,\qquad 2.65^2=7.0225.$$

So $2.64<\sqrt{7}<2.65$. To round to the nearest hundredth, compare with the midpoint, $2.645$:

$$2.645^2=6.996025<7.$$

Therefore $\sqrt{7}>2.645$, so it rounds to 2.65 to the nearest hundredth. This bracketing method is often the quickest approach when you need only a few decimal places.

Use square-root long division for more digits

The digit-by-digit method gives decimal digits in sequence. Split the number into pairs of digits, starting at the decimal point and moving outward. For example, write $10$ as $10\mid 00\mid 00\mid 00\dots$. Each pair you bring down supplies the next digit of the root.

At each stage, let $p$ be the root digits found so far, treated as a whole number. Bring down the next pair to form a new number $N$. Choose the largest digit $x$ from 0 to 9 such that

$$(20p+x)x\leq N.$$

Subtract $(20p+x)x$ from $N$, and append $x$ to the root digits. Repeat. After the first integer digits, each new root digit is a digit after the decimal point.

Worked example: find $\sqrt{10}$

The first group is 10. The largest square not greater than 10 is $3^2=9$, so the first root digit is 3 and the remainder is 1. Bring down the next pair, 00, to make 100. With $p=3$, the trial digit must satisfy $(60+x)x\leq100$. The largest choice is $x=1$, since $61\leq100$ but $124>100$. The root is now 3.1, and the remainder is 39.

Continue the same process:

Pair brought down	Number to test, $N$	Trial digit $x$	Product $(20p+x)x$	Root so far
00	3,900	6	3,756	3.16
00	14,400	2	12,644	3.162
00	175,600	2	126,484	3.1622
00	4,911,600	7	4,427,129	3.16227

For example, in the first row the current digits are $p=31$, so the test is $(620+x)x\leq3{,}900$. The largest valid digit is 6. The long-division result gives $\sqrt{10}\approx3.16227$, which rounds to 3.1623 to four decimal places.

Common mistakes

• Guessing from the digits inside the radical. The digits do not convert one at a time: $\sqrt{10}$ is not 3.1 just because 10 is close to 9.
• Forgetting to group digits in pairs. In the long-division method, groups are pairs on both sides of the decimal point; after the last given decimal pair, append pairs of zeros to continue.
• Using a too-wide bracket to round. Knowing a root is between 2.64 and 2.65 does not by itself settle rounding to the nearest hundredth; compare with the midpoint, 2.645.
• Treating an approximation as an exact decimal. A rounded value such as 3.1623 is not equal to $\sqrt{10}$; it is an approximation to a stated number of places.

For cube roots and other radicals, the same general idea applies: bracket the answer between nearby whole numbers whose corresponding powers lie on either side, then test decimal candidates by raising them to that power. The square-root long-division rule above is specifically for square roots.