Perfect Squares and Square Roots

What are perfect squares and square roots?

A perfect square is a nonnegative integer that equals an integer multiplied by itself. For example, \(49=7\times7=7^2\), so 49 is a perfect square.

For a nonnegative real number \(a\), the principal square root \(\sqrt a\) is the unique nonnegative number whose square is \(a\). Thus \(\sqrt{49}=7\). The symbol \(\sqrt{\phantom a}\) is the radical sign; the number or expression under it is the radicand.

Taking a square root reverses squaring when working with nonnegative numbers. This lesson explains that relationship, the role of signs, exact and approximate roots, and square-area applications.

Squaring is multiplication, not doubling

The exponent 2 means multiply the base by itself:

\[8^2=8\times8=64,\qquad 2\times8=16.\]

These operations are different. A negative number also has a nonnegative square: \((-8)^2=64\), because two negative factors give a positive product.

In contrast, \(-8^2=-(8^2)=-64\): the exponent is evaluated before the leading minus unless parentheses include the negative sign in the base.

A useful table of perfect squares

Each positive perfect square has both a positive and a negative integer square root, but the radical symbol selects only the nonnegative one. Zero has just one square root: zero.

See the connection between area and side length

For a square of side length \(s\ge0\), its area is \(A=s^2\). If the area is known, the side length is \(s=\sqrt A\).

The area is measured in square units, while the side is measured in length units. Dividing the area by 2 does not recover the side length.

Worked example: radical value versus equation solutions

Problem: Evaluate \(\sqrt{81}\) and solve \(x^2=81\).

Since \(9^2=81\) and 9 is nonnegative, \(\sqrt{81}=9\), not \(\pm9\).

For the equation, both \(9^2=81\) and \((-9)^2=81\). Therefore the solutions are

\[x=9\quad\text{or}\quad x=-9,\]

written compactly as \(x=\pm9\). The equation asks for every number whose square is 81; the radical asks for the principal one.

Also distinguish \(-\sqrt{81}=-9\), which places a minus outside the radical, from \(\sqrt{-81}\), whose radicand is negative.

Negative radicands and the real-number domain

The square of any real number is nonnegative. Consequently, \(\sqrt{-81}\) is not a real number, and \(x^2=-81\) has no real solution.

This is a restriction to real numbers, not a claim that negative radicands are never studied. Complex numbers extend the number system later. Here all square-root evaluations and estimates are within the real-number system.

Worked example: exact roots of fractions and decimals

Problem: Evaluate \(\sqrt{\frac{49}{100}}\) and \(\sqrt{0.0081}\).

Since \((\frac7{10})^2=\frac{49}{100}\) and \(\frac7{10}\) is nonnegative,

\[\sqrt{\frac{49}{100}}=\frac7{10}=0.7.\]

Similarly, \(0.09^2=0.0081\), so \(\sqrt{0.0081}=0.09\). Check the decimal placement by writing \(0.09=\frac9{100}\); its square is \(\frac{81}{10000}\).

The roots are exact. Although 0.0081 is not an integer perfect square, it is the square of a terminating decimal. The phrase “perfect square” in this lesson's integer table refers specifically to integers.

Why a nonperfect integer root does not terminate

The square root of a nonnegative integer that is not a perfect square is irrational: it cannot be written as a ratio of integers. Its decimal representation neither terminates nor eventually repeats.

For example, \(\sqrt{30}=5.477225\ldots\), with no endlessly repeating block. A calculator shows only a finite approximation; that display does not make the exact number rational.

This differs from \(\frac13=0.\overline3\), whose repeating decimal represents an exact rational number. A proof of the integer-root result belongs to later number-system work; you can use nearby squares here to estimate without assuming a finite decimal is exact.

Worked example: find a square's side and perimeter

Problem: A square patio has area 64 square metres. Find its side length and perimeter.

The side length is \(s=\sqrt{64}=8\) metres. Its perimeter is the total length of four sides:

\[P=4s=4\times8=32\text{ metres}.\]

Check: \(8\text{ m}\times8\text{ m}=64\text{ m}^2\). A physical side length cannot be negative, even though the equation \(s^2=64\), without that physical restriction, has two real solutions.

For a square of area 30 square metres, the exact side length would instead be \(\sqrt{30}\) metres, approximately 5.48 metres.

Optional extension: remove a perfect-square factor

For \(a,b\ge0\), \(\sqrt{ab}=\sqrt a\,\sqrt b\). Also, for \(a\ge0\) and \(b>0\), \(\sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}\). These real-number conditions matter.

Problem: Simplify \(\sqrt{45}\).

Factor \(45=9\times5\):

\[\sqrt{45}=\sqrt9\sqrt5=3\sqrt5.\]

The expression \(3\sqrt5\) is exact. It is approximately 6.71, but rounding does not replace the exact simplified form.

Do not use the product rule for addition. For instance, \(\sqrt{9+16}=\sqrt{25}=5\), while \(\sqrt9+\sqrt{16}=3+4=7\). The two expressions are different.

A squared variable needs an absolute value

For every real \(x\),

\[\sqrt{x^2}=|x|.\]

The absolute value is needed because the principal root must be nonnegative. If \(x=-6\), then \(\sqrt{x^2}=\sqrt{36}=6\), not \(-6\). When \(x\ge0\), the formula reduces to \(\sqrt{x^2}=x\).

In contrast, \((\sqrt a)^2=a\) for \(a\ge0\). Keep track of which operation is applied first and which sign restrictions are present.

Order of operations inside a radical

Treat the radical bar as grouping its entire radicand. For example,

\[\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5.\]

First evaluate exponents, then add inside the radical, then take the root. This is not \(3+4=7\). In an expression such as \(2\sqrt{36}+1\), take the root, multiply, then add: \(2\times6+1=13\).

Common mistakes

• Doubling instead of squaring: \(6^2=36\), not 12.
• Writing two values for the radical: \(\sqrt{49}=7\); \(x^2=49\) has solutions \(\pm7\).
• Ignoring the radical's domain: A negative radicand has no real square root.
• Forgetting parentheses around a negative base: \((-5)^2=25\), but \(-5^2=-25\).
• Distributing roots over sums: \(\sqrt{a+b}\) generally does not equal \(\sqrt a+\sqrt b\).
• Removing squares without checking signs: \(\sqrt{x^2}=|x|\), not always \(x\).
• Confusing exact and rounded values: Use \(\approx\) for estimates such as \(\sqrt{30}\approx5.48\).
• Mixing area and length units: A root of a square's area gives a side length, not an area or a perimeter.

Check your understanding

Try these before reading the explanations. These lesson self-checks do not change the separate Quizzes or Practice tabs.

1. Find \(13^2\) and \(\sqrt{169}\).
2. Evaluate \(\sqrt0\) and \(\sqrt{144}\).
3. Solve \(x^2=64\). How is the answer different from \(\sqrt{64}\)?
4. Evaluate \((-7)^2\) and \(-7^2\).
5. Does \(\sqrt{-25}\) have a real value? Evaluate \(-\sqrt{25}\).
6. Evaluate \(\sqrt{\frac{81}{100}}\) and \(\sqrt{0.0144}\).
7. Place \(\sqrt{70}\) between consecutive whole numbers.
8. Approximate \(\sqrt{70}\) to the nearest hundredth.
9. A square has area 100 square centimetres. Find its side length and perimeter.
10. Simplify \(\sqrt{75}\) using a perfect-square factor.
11. Evaluate \(\sqrt{x^2}\) when \(x=-4\).
12. Compare \(\sqrt{16+9}\) with \(\sqrt{16}+\sqrt9\).

Answers and explanations

1. \(13^2=169\) and \(\sqrt{169}=13\): check by squaring 13.
2. 0 and 12, respectively. The principal values are nonnegative.
3. \(x=8\) or \(x=-8\), whereas \(\sqrt{64}=8\) is one principal value.
4. 49 and \(-49\), respectively. Only the parenthesized expression includes the negative sign in the squared base.
5. \(\sqrt{-25}\) has no real value; \(-\sqrt{25}=-5\).
6. \(\frac9{10}=0.9\) and 0.12. Check: \(0.12^2=0.0144\).
7. \(8<\sqrt{70}<9\), since \(64<70<81\).
8. \(\sqrt{70}\approx8.37\). It lies between 8.36 and 8.37, and \(8.365^2=69.973225<70\), so it rounds upward.
9. Side length 10 centimetres; perimeter 40 centimetres. Check the area as \(10\times10=100\) square centimetres.
10. \(\sqrt{25\times3}=5\sqrt3\), an exact value.
11. \(\sqrt{16}=4=|-4|\), not \(-4\).
12. The first is 5 and the second is 7. Addition inside a radical differs from addition of two roots.

Common mistake

The square-root symbol selects the nonnegative root; solving a squared equation may require both signs.

Related lessons

• Exponents and Order of Operations: squared bases and grouping
• The Number Line and Absolute Value: signs and nonnegative distance
• Calculating with Decimals: square candidates and round estimates
• Converting Fractions and Decimals: rational forms and approximations

Sources and authorship

The public textbook resources below were consulted for topic coverage and mathematical checks. This lesson's explanations, application settings, self-checks, table, and diagram were independently created; textbook prose, exercise sets, and artwork are not reproduced. Public access does not imply unrestricted reuse: check each source's stated license before reproducing material.

• OpenStax Prealgebra 2e, 5.7: Simplify and Use Square Roots
• OpenStax Prealgebra 2e: preface and reuse conditions
• Siyavula Grade 8 Mathematics: Perfect Squares and Square Roots
• Siyavula: open-textbook access and licensing