Powers and square roots

When a number is multiplied by itself several times, it can be written in short – as a power.

A power

\displaystyle 2^5

is not

\displaystyle 2\cdot5

\displaystyle {2^5}={32}

, and

\displaystyle {2\cdot5}={10}

$\displaystyle {2\cdot2\cdot2\cdot2\cdot2}={2^5}$ . The number being multiplied ( $\displaystyle 2$ ) is the base, and the number of times it appears ( $\displaystyle 5$ ) is the exponent. Read it: “two to the power of five”. $\displaystyle {2^5}={32}$ .

A power with exponent 2 is a square: $\displaystyle {5^2}={25}$ – “five squared”, like the area of a square with side 5. A power with exponent 3 is a cube: $\displaystyle {5^3}={125}$ , like the volume of a cube with edge 5. And a number to the power of 1 is the number itself: $\displaystyle {7^1}={7}$ .

Exercise 1

Calculate.
First write the power as a product.

/ – 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

A negative base

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

, but

\displaystyle {-3^2}={-9}

A negative number can be raised to a power too: $\displaystyle {(-2)^3}={(-2)\cdot(-2)\cdot(-2)}={-8}$ , and $\displaystyle {(-2)^4}={16}$ . By the sign rule: when the exponent is even, the power is positive, and when it is odd – negative.

Mind the brackets. In $\displaystyle (-3)^2$ the base is $\displaystyle -3$ : $\displaystyle {(-3)^2}={(-3)\cdot(-3)}={9}$ . In $\displaystyle -3^2$ the base is only $\displaystyle 3$ : square 3, then take the opposite – $\displaystyle {-3^2}={-(3\cdot3)}={-9}$ .

Exercise 2

Calculate.
What is the base in each 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

Order of operations

Powers come after brackets, but before multiplication and division (and so before addition and subtraction too): $\displaystyle {2\cdot3^2}={2\cdot9}={18}$ , while $\displaystyle {(2\cdot3)^2}={6^2}={36}$ .

Exercise 3

Calculate.

/ – 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 square root

\displaystyle {\sqrt{25}}={5}

: a square root is always positive or 0.

$\displaystyle {5^2}={25}$ , and $\displaystyle {(-5)^2}={25}$ too. The square root of 25 is the positive number whose square is 25: $\displaystyle {\sqrt{25}}={5}$ . Read it: “the square root of 25”. More examples: $\displaystyle {\sqrt{100}}={10}$ , $\displaystyle {\sqrt{1}}={1}$ , $\displaystyle {\sqrt{0}}={0}$ .

The square of any number is positive or 0, so no number has the square $\displaystyle -25$ : $\displaystyle \sqrt{-25}$ does not exist.

Exercise 4

Calculate: $\displaystyle \sqrt{49}$ , $\displaystyle \sqrt{144}$ , $\displaystyle \sqrt{400}$ .
And now $\displaystyle \sqrt{9}+\sqrt{16}$ and $\displaystyle \sqrt{9+16}$ – are they equal?

/ – 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 5

$\displaystyle \sqrt{50}$ is not a whole number.
Between which two consecutive whole numbers is it?
And $\displaystyle \sqrt{10}$ ? Explain with squares.

Exercise 6

The area of a square is 81 cm².
How long is its side?
And the volume of a cube is 64 cm³ – how long is its edge?

The square’s side

/ – 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 cube’s edge

/ – 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 7

Open as a page

Adding and subtracting signed numbers

The sum of two opposite numbers
$\displaystyle (-a)+a$ $\displaystyle =$ $\displaystyle 0$
Subtracting a number is adding its opposite
$\displaystyle a-b$ $\displaystyle =$ $\displaystyle a+(-b)$
Subtracting a negative number
$\displaystyle a-(-b)$ $\displaystyle =$ $\displaystyle a+b$

Multiplying and dividing signed numbers

Positive times negative – negative
$\displaystyle a\cdot(-b)$ $\displaystyle =$ $\displaystyle -(a\cdot b)$
Negative times negative – positive
$\displaystyle (-a)\cdot(-b)$ $\displaystyle =$ $\displaystyle a\cdot b$
Dividing: different signs – the quotient is negative
$\displaystyle (-a):b$ $\displaystyle =$ $\displaystyle -(a:b)$ $\displaystyle b\neq0$
Dividing: the same sign – the quotient is positive
$\displaystyle (-a):(-b)$ $\displaystyle =$ $\displaystyle a:b$ $\displaystyle b\neq0$

Powers and roots

A power – $\displaystyle a$ the base, $\displaystyle n$ the exponent: how many times $\displaystyle a$ is multiplied
$\displaystyle a^{n}$ $\displaystyle =$ $\displaystyle \underbrace{a\cdot a\cdot\ldots\cdot a}_{n}$
A number to the power 1 is the number itself
$\displaystyle a^{1}$ $\displaystyle =$ $\displaystyle a$
A negative base with an even exponent – the power is positive
$\displaystyle (-a)^{2}$ $\displaystyle =$ $\displaystyle a^{2}$
The square root – the positive number (or 0) whose square is the number under the root
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The laws of arithmetic and like terms

The commutative law of addition
$\displaystyle a+b$ $\displaystyle =$ $\displaystyle b+a$
The commutative law of multiplication
$\displaystyle a\cdot b$ $\displaystyle =$ $\displaystyle b\cdot a$
The associative law of addition
$\displaystyle (a+b)+c$ $\displaystyle =$ $\displaystyle a+(b+c)$
The associative law of multiplication
$\displaystyle (a\cdot b)\cdot c$ $\displaystyle =$ $\displaystyle a\cdot(b\cdot c)$
A number times itself – squared
$\displaystyle x\cdot x$ $\displaystyle =$ $\displaystyle x^{2}$
Collecting like terms – add the coefficients; the letters stay
$\displaystyle ax+bx$ $\displaystyle =$ $\displaystyle (a+b)x$

The distributive law

The distributive law
$\displaystyle a(b+c)$ $\displaystyle =$ $\displaystyle ab+ac$
The distributive law over a difference
$\displaystyle a(b-c)$ $\displaystyle =$ $\displaystyle ab-ac$
A minus before brackets – every term inside changes its sign
$\displaystyle -(a-b)$ $\displaystyle =$ $\displaystyle -a+b$

Distance, speed and time

$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$