J. Richard's copy of Laws of Indices

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J. Richard Snape 5 years, 8 months ago

Created this as a copy of Laws of Indices.

Name	Status	Author	Last Modified
Laws of Indices	Ready to use	Elliott Fletcher	01/04/2020 09:09
Katherine's copy of Laws of Indices	draft	Katherine Tomlinson	01/08/2017 12:15
Eva's copy of Laws of Indices	draft	Eva Szatmari	12/06/2018 15:50
Laws of Indices [L6] Randomised	Doesn't work	Matthew James Sykes	25/07/2018 10:28
1. Laws of indices	draft	emma rand	27/07/2018 13:08
Ed's copy of Laws of Indices	draft	Ed Southwood	08/08/2022 10:18
Simon's copy of Laws of Indices	draft	Simon Thomas	04/03/2019 13:47
Blathnaid's copy of Simon's copy of Laws of Indices	draft	Blathnaid Sheridan	04/11/2019 13:00
David's copy of Laws of Indices	draft	David Goulding	11/03/2019 17:43
John's copy of Laws of Indices	draft	John Wick	17/06/2019 10:01
J. Richard's copy of Laws of Indices	draft	J. Richard Snape	15/08/2019 17:17
J. Richard's copy of Laws of Indices	draft	J. Richard Snape	15/08/2019 17:19
rekenregels van machten	draft	Johan Maertens	20/08/2019 16:36
Laws of Indeces 2	draft	Leonardo Juliano	07/10/2019 09:50
Ugur's copy of Laws of Indices	Ready to use	Ugur Efem	08/11/2021 16:45

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Question statement

Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

x	integer	6
y	integer	3
c	integer	4
d	integer	4
p	integer	7
q	integer	3
b	integer	8
g	integer	4

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..9 except y)

JME function reference

Description

Used in parts a,c and e

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
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1. Step Information only
Part b) Gap-fill
1. Gap Mathematical expression
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Part c) Gap-fill
1. Gap Mathematical expression
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Part d) Gap-fill
1. Gap Mathematical expression
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1. Step Information only
Part e) Gap-fill
1. Gap Mathematical expression
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Part f) Gap-fill
1. Gap Mathematical expression
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Write $a^{\var{x}} \times a^{\var{y}}$ as a single power of $a$ .

$a^{\var{x}} \times a^{\var{y}} =$ Unnamed gap.

Advice

a)

Here we are using the rule of indices: $a^m \times a^n = a^{m+n}$ .

Using this rule,

$\begin{align} a^\var{x} \times a^\var{y}\ &= a^\simplify[all, !collectNumbers]{{x}+{y}}\\ &= a^\var{x+y}. \end{align}$

b)

We are asked to find $\var{c}a^\var{p} \times \var{d}a^\var{q}$ .

Notice there is a constant in front of each of the terms.

To do this, write the product out explicitly, as

$\var{c}a^\var{p} \times \var{d}a^\var{q} = \var{c} \times \var{d} \times a^\var{p} \times a^\var{q}.$

We know that $\var{c} \times \var{d} = \var{c*d}$ , and using the rule of indices: $a^\var{p} \times a^\var{q} = a^\var{p+q}$ .

Therefore:

$\begin{align} \var{c}a^\var{p} \times \var{d}a^\var{q}&= \var{c*d} \times a^\var{p+q} \\ &= \simplify{{c*d}*a^{p+q}}. \end{align}$

c)

Here we are using: $a^m \div a^n = a^{m-n}$ .

We are asked to simplify the expression, $\displaystyle\simplify{{b}*a^{x}/({g}*a^{y})}$ .

To do this, we just have to use the previously mentioned rule of indices. We write this out explicity as

$\simplify{{b}*a^{x}/({g}*a^{y})} = \simplify{{b}/{g}} \times \simplify{a^{x}/(a^{y})}.$

Using rules of indices,

$\begin{align} \frac{a^\var{x}}{a^\var{y}} &= a^\var{x} \div a^\var{y}\\ &= a^\simplify[all, !collectNumbers]{{x}-{y}}\\ &= a^\var{x-y}. \end{align}$

Therefore,

$\begin{align} \frac{\var{b}a^\var{x}}{\var{g}a^\var{y}} &= \simplify{{b}/{g}} \times \simplify{a^{{x}-{y}}}\\ &= \simplify{{b}/{g}*a^{x-y}}. \end{align}$

Alternatively,

Using the rule of indices: $a^{-m} = \displaystyle\frac{1}{a^{m}}$ , we can rewrite the question as:

$\begin{align} \frac{\var{b}a^\var{x}}{\var{g}a^\var{y}} &= \simplify{{b}/{g}} \times \frac{a^\var{x}}{a^\var{y}}\\ &= \simplify{{b}/{g}} \times a^\var{x} \times a^{-\var{y}}. \end{align}$

And then using the rule: $a^m \times a^n = a^{m+n}$ , this becomes:

$\begin{align} \simplify{{b}/{g}} \times a^\var{x} \times a^{-\var{y}} &= \simplify{{b}/{g}} \times a^\simplify[all,!collectNumbers]{{x}+(-{y})}\\ &= \simplify{{b}/{g}*a^{x-y}}. \end{align}$

d)

The question asks us to simplify $(\simplify{{c}*a^{p}})^{\var{q}}$ .

To do this we use the rules:

$(a^{m})^{n} = a^{mn},$

$(ab)^m = a^mb^m.$

We can then expand the equation as

$(\simplify{{c}*a^{p}})^{\var{q}}= \var{c}^{\var{q}} \times (a^{\var{p}})^{\var{q}}.$

Then using the rule of indices mentioned previously,

$\begin{align} (\simplify{{c}*a^{p}})^{\var{q}}&= \simplify{{c}^{q}} \times a^\var{p*q}\\ &= \simplify{{c}^{q}*a^{p*q}}. \end{align}$

e)

The question asks us to simplify $\sqrt[\var{d}]{\var{x}^\var{d}a}$ .

To do this we use the rules:

$a^\frac{1}{m} = \sqrt[m]{a},$

$(ab)^m = a^mb^m.$

We can expand the expression as follows:

$\begin{align} \sqrt[\var{d}]{a} &= (\simplify{a})^\frac{1}{\var{d}}\\ &= a^\frac{1}{\var{d}}. \end{align}$

f)

The question requires us to simplify $\sqrt[\var{c}]{a^\var{q}}$ .

Here, we use the rule of indices: $a^\frac{n}{m} = \sqrt[m]{a^n}$ , allowing us to expand the expression as follows:

$\begin{align} \sqrt[\var{c}]{\simplify{a^{q}}} &= \simplify[fractionnumbers,all]{(a^{q})^{{1}/{{c}}}}\\ &= \simplify[fractionnumbers,all]{a^{{q}/{c}}}. \end{align}$

Give a worked solution to the whole question.

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J. Richard Snape 5 years, 8 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

e)

f)

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J. Richard's copy of Laws of Indices

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

J. Richard Snape 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

e)

f)

Generated value: `integer`