Practice: powers and roots

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

Calculate.
Mind the order of operations.

/ – 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

Calculate: $\displaystyle \sqrt{64}$ , $\displaystyle \sqrt{121}$ , $\displaystyle \sqrt{900}$ , $\displaystyle \sqrt{100}-\sqrt{36}$ , $\displaystyle \sqrt{100-36}$ , $\displaystyle \left(\sqrt{7}\right)^2$ .

/ – 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

Write 64 as a power in three different ways.
And 81 – in how many ways can it be done?

Without calculating: is the number positive or negative? Explain.
$\displaystyle (-7)^{15}$ , $\displaystyle \quad(-7)^{20}$ , $\displaystyle \quad-7^{20}$ , $\displaystyle \quad(-1)^{100}\cdot(-3)$ .

Which number is missing from the box?
$\displaystyle {\square^2}={49}$ (there are two possibilities!), $\displaystyle {\quad\square^3}={-27}$ , $\displaystyle {\quad2^{\square}}={128}$ , $\displaystyle {\quad(-10)^{\square}}={10000}$ .

/ – 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

Which is greater: $\displaystyle 2^{10}$ or $\displaystyle 10^2$ ? $\displaystyle \;3^4$ or $\displaystyle 4^3$ ? $\displaystyle \;(-2)^3$ or $\displaystyle (-3)^2$ ? Explain.

The area of a square is 169 cm².
What is its perimeter?

/ – 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
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$