The laws of powers

The laws of powers let you calculate with powers without writing out all their factors. A reminder from seventh grade: a power is a product of equal factors.

$\displaystyle {2^5}={2\cdot2\cdot2\cdot2\cdot2}={32}$
The number being multiplied ( $\displaystyle 2$ ) is the base; the number of times it appears ( $\displaystyle 5$ ) is the exponent.
The laws hold for numbers and for variables alike. The exponents here are whole numbers – positive, or 0.

Multiplying powers with the same base

The exponents are added, not multiplied:

\displaystyle {x^3\cdot x^4}={x^7}

, not

\displaystyle x^{12}

When you multiply powers with the same base, the base stays and the exponents are added:

\displaystyle a^m\cdot a^n

\displaystyle =

\displaystyle a^{m+n}

Why: $\displaystyle {2^3\cdot2^4}={(2\cdot2\cdot2)\cdot(2\cdot2\cdot2\cdot2)}={2^7}$ – three factors and four more.
$\displaystyle {x^5\cdot x^2}={x^7}$ ; $\displaystyle {\;x\cdot x^4}={x^5}$ (a variable on its own is a power with exponent 1: $\displaystyle {x}={x^1}$ ).
Powers with different bases stay as they are: $\displaystyle 2^3\cdot3^4$ .

Dividing powers with the same base

The exponents are subtracted, not divided:

\displaystyle {x^9:x^3}={x^6}

, not

\displaystyle x^3

When you divide powers with the same base, the exponents are subtracted:

\displaystyle a^m:a^n=\frac{a^m}{a^n}=a^{m-n}

\displaystyle a\neq0

Why: $\displaystyle {\dfrac{5^6}{5^2}}={\dfrac{5\cdot5\cdot5\cdot5\cdot5\cdot5}{5\cdot5}}={5^4}$ – two of the six factors cancel out.
$\displaystyle {x^9:x^3}={x^6}$ ; $\displaystyle {\;\dfrac{y^7}{y}}={y^6}$ .

Exercise 1

Write as a single power, or 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

A power of a power

Compare:

\displaystyle {(x^2)^3}={x^6}

, but

\displaystyle {x^2\cdot x^3}={x^5}

When a power is raised to a power, the exponents are multiplied:

\displaystyle (a^m)^n

\displaystyle =

\displaystyle a^{m\cdot n}

Why: $\displaystyle {(2^3)^2}={2^3\cdot2^3}={2^6}$ – twice three factors.
$\displaystyle {(x^4)^3}={x^{12}}$ ; $\displaystyle {\;(10^2)^5}={10^{10}}$ .

A power of a product

The power applies to every factor in the brackets:

\displaystyle {(2x)^3}={8x^3}

, not

\displaystyle 2x^3

A product raised to a power is the product of the powers of its factors:

\displaystyle (ab)^n

\displaystyle =

\displaystyle a^n\cdot b^n

Why: $\displaystyle {(2x)^3}={2x\cdot2x\cdot2x}={2\cdot2\cdot2\cdot x\cdot x\cdot x}={8x^3}$ .
$\displaystyle {(3a^2)^2}={3^2\cdot(a^2)^2}={9a^4}$ ; $\displaystyle {\;(-xy)^3}={-x^3y^3}$ .

A power of a fraction

A fraction raised to a power is the power of the numerator over the power of the denominator:

\displaystyle \left(\frac{a}{b}\right)^n

\displaystyle =

\displaystyle \frac{a^n}{b^n}

\displaystyle b\neq0

$\displaystyle {\left(\dfrac{x}{2}\right)^3}={\dfrac{x^3}{8}}$ ; $\displaystyle {\;\left(\dfrac{2}{3}\right)^2}={\dfrac{4}{9}}$ .

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

The exponent 0

\displaystyle {3x^0}={3\cdot1}={3}

, but

\displaystyle {(3x)^0}={1}

Any number other than 0, to the power of 0, is 1:

\displaystyle a^0

\displaystyle =

\displaystyle 1

\displaystyle a\neq0

Why: $\displaystyle {\dfrac{5^3}{5^3}}={1}$ (a number divided by itself), and by the law of division $\displaystyle {\dfrac{5^3}{5^3}}={5^{3-3}}={5^0}$ . So $\displaystyle 5^0$ has to be 1 for the law to stay true.
$\displaystyle {7^0}={1}$ ; $\displaystyle {\;(-3)^0}={1}$ ; $\displaystyle {\;x^0}={1}$ .
$\displaystyle 0^0$ has no meaning.

The laws together

In a product of terms, multiply the numbers with each other, and the powers of each variable by the laws: $\displaystyle {3x^2\cdot5x^4}={15x^6}$ ; $\displaystyle {\;(2x^3)^2\cdot x}={4x^6\cdot x}={4x^7}$ ; $\displaystyle {\;\dfrac{x^5\cdot x^3}{x^4}}={\dfrac{x^8}{x^4}}={x^4}$ .

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

Calculate without a calculator.
In b and c the exponents are equal – use the laws backwards: $\displaystyle {a^n\cdot b^n}={(ab)^n}$ , $\displaystyle {\;\dfrac{a^n}{b^n}}={\left(\dfrac{a}{b}\right)^n}$ .

/ – 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

Noa wrote: $\displaystyle {x^3\cdot x^4}={x^{12}}$ and $\displaystyle {(2x)^3}={2x^3}$ .
What is wrong in each?
Explain, and simplify correctly.

/ – 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 $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$