Exercises: Power Laws

Conclude as far as possible.

$a^{3} : a^{6}$

For this task you need the following basic knowledge: Powers
First Representation
$\frac{a \cdot a \cdot a}{a \cdot a \cdot a \cdot a \cdot a \cdot a}$ $=$
↓
Shorten by cancelling three factors of $a$ .
$=$ $\frac{1}{a \cdot a \cdot a}$
$=$ $\frac{1}{a^{3}} = a^{- 3}$
Second Representation
$a^{3} : a^{6}$ $=$
↓
Apply the power laws.
$=$ $a^{3 - 6}$
$=$ $a^{- 3}$
$2 x^{- 2} \cdot 3 x^{3}$

For this task you need the following basic knowledge: Powers
Power laws
$2 x^{- 2} \cdot 3 x^{3}$ $=$
↓
Use the commutative law to group numbers and variables together.
$=$ $2 \cdot 3 \cdot x^{- 2} \cdot x^{3}$
↓
Apply the power laws.
$=$ $2 \cdot 3 \cdot x^{- 2 + 3}$
$=$ $6 x$
$10^{- 12} : 10^{- 3}$

For this task you need the following basic knowledge: Powers
Power laws
$10^{- 12} : 10^{- 3}$ $=$
↓
Apply the power laws.
$=$ $10^{- 12 - (- 3)}$
$=$ $10^{- 9}$
$6 : 2^{3} - 9 \cdot 3^{- 2}$

For this task you need the following basic knowledge: Powers
$6 : 2^{3} - 9 \cdot 3^{- 2}$ $=$
↓
Write $9$ as $3^{2}$ .
$=$ $6 : 2^{3} - 3^{2} \cdot 3^{- 2}$
↓
Apply the power laws.
$=$ $6 : 8 - 3^{2 + (- 2)}$
$=$ $6 : 8 - 3^{0}$
$=$ $0.75 - 1$
$=$ $- 0.25$
$x^{- n} \cdot x$

For this task you need the following basic knowledge: Powers
$x^{- n} \cdot x$ $=$
↓
Write $x$ as $x^{1}$ .
$=$ $x^{- n} \cdot x^{1}$
↓
Apply the power laws.
$=$ $x^{- n + 1}$
Alternative solution
$x^{- n} \cdot x$ $=$
↓
Write $x^{- n}$ as $\frac{1}{x^{n}}$ .
$=$ $\frac{1}{x^{n}} \cdot x^{1}$
$=$ $\frac{x^{1}}{x^{n}}$
↓
Apply the power laws.
$=$ $x^{1 - n}$
$0.5 x^{2} + 1.5 x^{3}$

For this task you need the following basic knowledge: Powers
This term cannot be simplified further, since two different powers occur. However, what one could do is factorizing the term:
$0.5 x^{2} + 1.5 x^{3}$ $=$
↓
Factorize $0.5 x^{2}$ .
$=$ $0.5 x^{2} (1 + 3 x)$
${(\frac{x^{3} y^{- 4}}{y^{- 5} y^{2}})}^{- 2}$

For this task you need the following basic knowledge: Powers
First apply the power laws. You may get the minus out of the exponent by "flipping" the fractions like $x^{- 2} = \frac{1}{x^{2}}$ .
${(\frac{x^{3} y^{- 4}}{y^{- 5} y^{2}})}^{- 2}$ $=$ ${(\frac{y^{- 5} y^{2}}{x^{3} y^{- 4}})}^{2}$
↓
Apply the power laws.
$=$ ${(\frac{y^{- 5 + 2}}{x^{3} y^{- 4}})}^{2}$
$=$ ${(\frac{y^{- 3}}{x^{3} y^{- 4}})}^{2}$
↓
Shorten by $y^{- 3}$ .
$=$ ${(\frac{1}{x^{3} y^{- 1}})}^{2}$
↓
Apply the power laws once more.
$=$ ${(\frac{y}{x^{3}})}^{2}$
$=$ $\frac{y^{2}}{x^{6}}$
${(2 x^{3})}^{2}$

For this task you need the following basic knowledge: Powers
${(2 x^{3})}^{2}$ $=$
↓
Apply the power law ${(a \cdot b)}^{x} = a^{x} \cdot b^{x}$
$=$ $2^{2} \cdot x^{3 \cdot 2}$
$=$ $4 \cdot x^{6}$

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$\frac{a \cdot a \cdot a}{a \cdot a \cdot a \cdot a \cdot a \cdot a}$	$=$
	↓	Shorten by cancelling three factors of $a$ .
	$=$	$\frac{1}{a \cdot a \cdot a}$
	$=$	$\frac{1}{a^{3}} = a^{- 3}$

$a^{3} : a^{6}$	$=$
	↓	Apply the power laws.
	$=$	$a^{3 - 6}$
	$=$	$a^{- 3}$

$2 x^{- 2} \cdot 3 x^{3}$	$=$
	↓	Use the commutative law to group numbers and variables together.
	$=$	$2 \cdot 3 \cdot x^{- 2} \cdot x^{3}$
	↓	Apply the power laws.
	$=$	$2 \cdot 3 \cdot x^{- 2 + 3}$
	$=$	$6 x$

$10^{- 12} : 10^{- 3}$	$=$
	↓	Apply the power laws.
	$=$	$10^{- 12 - (- 3)}$
	$=$	$10^{- 9}$

$6 : 2^{3} - 9 \cdot 3^{- 2}$	$=$
	↓	Write $9$ as $3^{2}$ .
	$=$	$6 : 2^{3} - 3^{2} \cdot 3^{- 2}$
	↓	Apply the power laws.
	$=$	$6 : 8 - 3^{2 + (- 2)}$
	$=$	$6 : 8 - 3^{0}$
	$=$	$0.75 - 1$
	$=$	$- 0.25$

$x^{- n} \cdot x$	$=$
	↓	Write $x$ as $x^{1}$ .
	$=$	$x^{- n} \cdot x^{1}$
	↓	Apply the power laws.
	$=$	$x^{- n + 1}$

$x^{- n} \cdot x$	$=$
	↓	Write $x^{- n}$ as $\frac{1}{x^{n}}$ .
	$=$	$\frac{1}{x^{n}} \cdot x^{1}$
	$=$	$\frac{x^{1}}{x^{n}}$
	↓	Apply the power laws.
	$=$	$x^{1 - n}$

$0.5 x^{2} + 1.5 x^{3}$	$=$
	↓	Factorize $0.5 x^{2}$ .
	$=$	$0.5 x^{2} (1 + 3 x)$

${(\frac{x^{3} y^{- 4}}{y^{- 5} y^{2}})}^{- 2}$	$=$	${(\frac{y^{- 5} y^{2}}{x^{3} y^{- 4}})}^{2}$
	↓	Apply the power laws.
	$=$	${(\frac{y^{- 5 + 2}}{x^{3} y^{- 4}})}^{2}$
	$=$	${(\frac{y^{- 3}}{x^{3} y^{- 4}})}^{2}$
	↓	Shorten by $y^{- 3}$ .
	$=$	${(\frac{1}{x^{3} y^{- 1}})}^{2}$
	↓	Apply the power laws once more.
	$=$	${(\frac{y}{x^{3}})}^{2}$
	$=$	$\frac{y^{2}}{x^{6}}$

${(2 x^{3})}^{2}$	$=$
	↓	Apply the power law ${(a \cdot b)}^{x} = a^{x} \cdot b^{x}$
	$=$	$2^{2} \cdot x^{3 \cdot 2}$
	$=$	$4 \cdot x^{6}$

Exercises: Power Laws

First Representation

Second Representation

Power laws

Power laws