The laws of square roots

The laws of square roots let you calculate with roots and write them in a simpler form. A reminder from seventh grade: the square root of a number $\displaystyle a\ge0$ is the number, positive or 0, whose square is $\displaystyle a$ .

$\displaystyle {\sqrt{49}}={7}$ , since $\displaystyle {7^2}={49}$ .
$\displaystyle \sqrt{-4}$ does not exist: the square of any number is positive or 0.
Most roots are not whole numbers: $\displaystyle \sqrt{2}$ is a little more than $\displaystyle 1.41$ .

The square of a root and the root of a square

\displaystyle {\sqrt{(-3)^2}}={\sqrt{9}}={3}

, not

\displaystyle -3

A root squared gives back the number under it:

\displaystyle \left(\sqrt{a}\right)^2

\displaystyle =

\displaystyle a

\displaystyle a\ge0

$\displaystyle {\left(\sqrt{7}\right)^2}={7}$ – by the definition of the root.
The other way round, $\displaystyle {\sqrt{5^2}}={\sqrt{25}}={5}$ . So $\displaystyle {\sqrt{a^2}}={a}$ when $\displaystyle a\ge0$ .
For a negative number the root of its square is the opposite number: $\displaystyle {\sqrt{(-5)^2}}={\sqrt{25}}={5}$ .

The root of a product

A sum has no such law:

\displaystyle {\sqrt{9+16}}={\sqrt{25}}={5}

, but

\displaystyle {\sqrt{9}+\sqrt{16}}={7}

The root of a product is the product of the roots:

\displaystyle \sqrt{ab}

\displaystyle =

\displaystyle \sqrt{a}\cdot\sqrt{b}

\displaystyle a\ge0,\ b\ge0

$\displaystyle {\sqrt{4\cdot9}}={\sqrt{36}}={6}$ , and $\displaystyle {\sqrt{4}\cdot\sqrt{9}}={2\cdot3}={6}$ .
Why: $\displaystyle {\left(\sqrt{a}\cdot\sqrt{b}\right)^2}={\left(\sqrt{a}\right)^2\cdot\left(\sqrt{b}\right)^2}={ab}$ by the law of a power of a product, and $\displaystyle \sqrt{a}\cdot\sqrt{b}$ is not negative.
Backwards, two roots become one: $\displaystyle {\sqrt{2}\cdot\sqrt{8}}={\sqrt{16}}={4}$ .

The root of a fraction

The root of a fraction is the root of the numerator over the root of the denominator:

\displaystyle \sqrt{\frac{a}{b}}

\displaystyle =

\displaystyle \frac{\sqrt{a}}{\sqrt{b}}

\displaystyle a\ge0,\ b>0

$\displaystyle {\sqrt{\dfrac{9}{25}}}={\dfrac{3}{5}}$
Backwards: $\displaystyle {\dfrac{\sqrt{50}}{\sqrt{2}}}={\sqrt{\dfrac{50}{2}}}={\sqrt{25}}={5}$ .

Exercise 1

Calculate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Simplifying a root

\displaystyle 2\sqrt{3}

means

\displaystyle 2\cdot\sqrt{3}

A root is simplified by taking a perfect square out from under it:

\sqrt{12}=\sqrt{4\cdot3}=\sqrt{4}\cdot\sqrt{3}=2\sqrt{3}

Look for the largest perfect square that divides the number: $\displaystyle {\sqrt{50}}={\sqrt{25\cdot2}}={5\sqrt{2}}$ .
With a smaller square you are not done yet: $\displaystyle {\sqrt{72}}={\sqrt{4\cdot18}}={2\sqrt{18}}={2\cdot3\sqrt{2}}={6\sqrt{2}}$ . With the largest one, in one step: $\displaystyle {\sqrt{72}}={\sqrt{36\cdot2}}={6\sqrt{2}}$ .
The perfect squares to look for: $\displaystyle 4, 9, 16, 25, 36, 49, 64, 81, 100$ .

Exercise 2

Simplify.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Adding like roots

\displaystyle \sqrt{2}+\sqrt{3}

is not

\displaystyle \sqrt{5}

\displaystyle {1.41+1.73}={3.14}

, and

\displaystyle \sqrt{5}

is about

\displaystyle 2.24

Roots with the same number under them are added like like terms: $\displaystyle {2x+5x}={7x}$ , and in the same way $\displaystyle {2\sqrt{3}+5\sqrt{3}}={7\sqrt{3}}$ .

$\displaystyle {6\sqrt{5}-\sqrt{5}}={5\sqrt{5}}$
Different roots stay as they are: $\displaystyle \sqrt{3}+\sqrt{5}$ .
Sometimes simplifying first makes them like: $\displaystyle {\sqrt{12}+\sqrt{27}}={2\sqrt{3}+3\sqrt{3}}={5\sqrt{3}}$ .

Exercise 3

Simplify.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The side of a square is 5 cm.
Find the length of its diagonal with the Pythagorean theorem, and write it as a simplified root.
The area of another square is 48 cm².
Find the length of its side, and write it as a simplified root.

The diagonal

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The side of the other square

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: algebra, grade 9

Open as a page

The laws of powers

Multiplying powers with the same base
$\displaystyle a^{m}\cdot a^{n}$ $\displaystyle =$ $\displaystyle a^{m+n}$
Dividing powers with the same base
$\displaystyle a^{m}:a^{n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^{m}\right)^{n}$ $\displaystyle =$ $\displaystyle a^{m\cdot n}$
A power of a product
$\displaystyle (a\cdot b)^{n}$ $\displaystyle =$ $\displaystyle a^{n}\cdot b^{n}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{n}$ $\displaystyle =$ $\displaystyle \frac{a^{n}}{b^{n}}$ $\displaystyle b\neq0$
The power zero
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$

The laws of square roots

The root of a product
$\displaystyle \sqrt{a\cdot b}$ $\displaystyle =$ $\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle a\ge0,\ b\ge0$
The root of a quotient
$\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle a\ge0,\ b>0$
The square of a root
$\displaystyle \left(\sqrt{a}\right)^{2}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$
The root of a square
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The extended distributive law and the short multiplication formulas

The extended distributive law
$\displaystyle (a+b)(c+d)$ $\displaystyle =$ $\displaystyle ac+ad+bc+bd$
The square of a sum
$\displaystyle (a+b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}+2ab+b^{2}$
The square of a difference
$\displaystyle (a-b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}-2ab+b^{2}$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^{2}-b^{2}$

The parabola

Standard form
$\displaystyle y=ax^{2}+bx+c$
Vertex form – the vertex is $\displaystyle (b,k)$
$\displaystyle y=a(x-b)^{2}+k$
Product form – it meets the $\displaystyle x$ -axis at $\displaystyle x_1$ and $\displaystyle x_2$
$\displaystyle y=a(x-x_1)(x-x_2)$
The vertex’s $\displaystyle x$ , in standard form
$\displaystyle x=-\frac{b}{2a}$

The quadratic equation

The roots of $\displaystyle {ax^{2}+bx+c}={0}$ , when $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$