Mathematics · Numbers and calculation · Grade 8

Roots: from squares to powers

What is the side of a square room with an area of 50 m²? Squares on a grid will not give it

The situation

What is the side of a square room with an area of 50 m²? 7 × 7 = 49 — too little, 8 × 8 = 64 — too much: the side lies between 7 and 8, but grid squares will not find it. And what about the edge of a cubic container of 1 litre? The root answers these questions: we find √50 by closing in on it, read ∛1000 from the cube, and then move on to the properties of roots and to powers.

Topic

Square and cube roots, the arithmetic square root and its properties, powers with a rational exponent. The side of a square room of area 50 m² and the edge of a cubic 1-litre container: roots, approximate values, a root as a power

Key ideas

Square and cube roots
Arithmetic root
a^(m/n) = ⁿ√aᵐ

Formulas

\sqrt{a} = b \Leftrightarrow b \geq 0, b^{2} = a (a \geq 0)

Arithmetic square root: the non-negative number whose square equals $a$ . ${(\sqrt{a})}^{2} = a$ , $\sqrt{a^{2}} = | a |$ .

\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}, \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} (a \geq 0, b > 0)

The root of a product and of a quotient; from this, taking a factor out from under the root: $\sqrt{50} = \sqrt{25 \cdot 2} = 5 \sqrt{2}$ .

\sqrt[3]{V} = a \Leftrightarrow a^{3} = V

Cube root: the edge of a cube of volume $V$ . $\sqrt[3]{1000} = 10$ , $\sqrt[3]{- 8} = - 2$ (for an odd index the root of a negative number exists).

a^{\frac{m}{n}} = \sqrt[n]{a^{m}} (a > 0)

Power with a rational exponent: $a^{1 / 2} = \sqrt{a}$ , $a^{1 / 3} = \sqrt[3]{a}$ ; the laws of powers ( $a^{p} a^{q} = a^{p + q}$ , ${(a^{p})}^{q} = a^{p q}$ ) still hold.

Theory

Square root — closing in on it

The side of a square of area $S$ is $\sqrt{S}$ (“the square root of $S$ ”): $\sqrt{49} = 7$ , because $7^{2} = 49$ . If $S = 50$ , the side is not a whole number: $7^{2} = 49 < 50 < 64 = 8^{2}$ , so $7 < \sqrt{50} < 8$ . We try tenths: ${7.0}^{2} = 49$ , ${7.1}^{2} = 50.41$ — so $7.0 < \sqrt{50} < 7.1$ ; hundredths: ${7.07}^{2} = 49.98$ , ${7.08}^{2} = 50.13$ . So $\sqrt{50} \approx 7.07$ — we can go on to any accuracy we need. $\sqrt{50}$ is not a rational number: its decimal expansion is infinite and does not repeat.

Cube root

The edge of a cube of volume $V$ is $\sqrt[3]{V}$ : $a^{3} = V$ . 1 litre $= 1000$ cm³, so the edge of a one-litre cubic container is $\sqrt[3]{1000} = 10$ cm. If we make the volume twice as large, the edge does not become twice as long, but $\sqrt[3]{2} \approx 1.26$ times longer: $\sqrt[3]{2000} \approx 12.6$ cm. A cube root can also be taken of a negative number: $\sqrt[3]{- 8} = - 2$ , because ${(- 2)}^{3} = - 8$ .

Arithmetic root

The equation $x^{2} = 4$ has two solutions: $2$ and $- 2$ . The non-negative one is the arithmetic square root: $\sqrt{4} = 2$ (never $\pm 2$ !). So $\sqrt{a}$ is a single number, and the graph $y = \sqrt{x}$ is the right half of the parabola reflected in the line $y = x$ . From the definition: ${(\sqrt{a})}^{2} = a$ ; the converse needs care: $\sqrt{a^{2}} = | a |$ , for example $\sqrt{{(- 4)}^{2}} = 4$ . The square root of a negative number does not exist among the real numbers.

Properties and taking a factor out

The area of a square with side $\sqrt{a} \cdot \sqrt{b}$ is ${(\sqrt{a})}^{2} {(\sqrt{b})}^{2} = a b$ , so its side is also $\sqrt{a b}$ : $\sqrt{a b} = \sqrt{a} \sqrt{b}$ ( $a, b \geq 0$ ). In the same way $\sqrt{a / b} = \sqrt{a} / \sqrt{b}$ ( $b > 0$ ). From this, taking a factor out from under the root: $\sqrt{50} = \sqrt{25 \cdot 2} = 5 \sqrt{2}$ — a square with side $5 \sqrt{2}$ consists of $25$ cells of $\sqrt{2} \times \sqrt{2}$ , each of area 2. The reverse is putting a factor under the root: $5 \sqrt{2} = \sqrt{50}$ . For a sum there is no such property: $\sqrt{9 + 16} = 5 \neq 3 + 4$ .

Power with a rational exponent

What is $a^{1 / 2}$ ? If the law of powers is to still hold, $a^{1 / 2} \cdot a^{1 / 2} = a^{1 / 2 + 1 / 2} = a$ — that is, $a^{1 / 2}$ is the positive number whose square is $a$ : $a^{1 / 2} = \sqrt{a}$ . In general $a^{m / n} = \sqrt[n]{a^{m}}$ ( $a > 0$ ): $8^{2 / 3} = \sqrt[3]{64} = 4$ , $2^{- 1 / 2} = 1 / \sqrt{2}$ . On a graph these points lie on the smooth curve $y = a^{x}$ , between the whole-number points. When simplifying expressions, the laws of powers work: ${(a^{1 / 2} a^{1 / 3})}^{6} = {(a^{5 / 6})}^{6} = a^{5}$ . In an irrational equation ( $\sqrt{x + 3} = x - 3$ ) both sides are squared — this may introduce an extraneous root, so the answer is checked.

About the model

Roots that are not whole numbers are shown rounded to two decimal places (“≈”); the exact value is written like $\sqrt{50}$ , $5 \sqrt{2}$ . The sliders are limited: area up to 100, volume up to 1000 cm³; the drawing of the square is always the same size — the scale changes with the numbers. A power with a rational exponent is considered only for a positive base; for a negative base ( ${(- 8)}^{1 / 3}$ ) no definition has been agreed, so it is not shown.

Did you know?

The root sign “√” first appeared in 1525 in the book “Die Coss” by the German mathematician Christoff Rudolff — it is thought to come from the letter “r” of the Latin word “radix” (root); the bar over the expression under the root was introduced by René Descartes in 1637.
The Babylonian clay tablet YBC 7289 (about 1800–1600 BC) gives the sexagesimal value of $\sqrt{2}$ on the diagonal of a square: 1;24,51,10 — in decimal this is 1.41421296…, correct to six decimal places.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Fractions: from pizza to formula — Grade 5
Prime numbers, HCF and LCM — Grade 7
The number line: from −7 to √2 — Grade 7
Powers: from folding to a formula — Grade 7