Solve equations which include a single even power (e.g. x^even=blah)

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Xiaodan Leng 5 years, 8 months ago

Created this as a copy of Solve equations which include a single even power (e.g. x^even=blah).

Name	Status	Author	Last Modified
Solve equations which include a single odd power (e.g. x^odd=blah)	Ready to use	Ben Brawn	20/03/2025 04:05
Solve equations which include a single even power (e.g. x^even=blah)	Ready to use	Ben Brawn	20/03/2025 04:14
Solve equations which include a single root (e.g. \sqrt{x}=blah)	Ready to use	Ben Brawn	21/02/2025 08:07
Eirik's copy of Solve equations which include a single root (e.g. \sqrt{x}=blah)	draft	Eirik Heen	14/06/2017 09:28
Hannah's copy of Solve equations which include a single root (e.g. \sqrt{x}=blah)	draft	Hannah Bartholomew	26/06/2018 17:17
Solve equations which include a single odd power (e.g. x^odd=blah)	draft	Xiaodan Leng	11/07/2019 02:28
Solve equations which include a single root (e.g. \sqrt{x}=blah)	draft	Xiaodan Leng	11/07/2019 02:28
Solve equations which include a single even power (e.g. x^even=blah)	draft	Xiaodan Leng	11/07/2019 02:29
Leonardo's copy of Solve equations which include a single odd power (e.g. x^odd=blah)	draft	Leonardo Juliano	30/08/2019 13:59

There are 3 other versions that do you not have access to.

Question statement

Please complete the following.

Give any introductory information the student needs.

			Name	Type	Generated Value

intpower	integer	2
intrhs	number	64
intsoln	integer	8

			Name	Type	Generated Value

bpower	integer	2
bsoln	integer	7
bxcoeff	integer	-2
bb	integer	81
bc	number	-17
brhs	number	49

			Name	Type	Generated Value

cpower	integer	6
cc	integer	-9
cb	integer	-57
cxcoeff	integer	5
csolnapprox	number	782757.7896959998

			Name	Type	Generated Value

dc	integer	2
db	integer	-50
ddenom	integer	4

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..8#2)

JME function reference

Description

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`

→ Used by:

intrhs
intsoln

Advice
"Part a)" - prompt

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part c) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part d) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

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

If $x^\var{intpower}=\var{intrhs}$ , then $x=$ Gap 0., or Gap 1..

(Please enter the smaller value in the first gap and the larger value on the second gap)

Advice

Suppose you wanted to solve $x^2=4$ , that is, you wanted the number that squares to give four. The number $2$ comes to mind, however, $-2$ is also a solution since $(-2)^2=(-2)\times(-2)=4$ . Recall the product of two negatives is a positive, so the product of any even number of negative numbers is positive. This means when we find a positive solution to an equation like $x^4=10\,000$ there will also be a negative solution, in particular, the solution to $x^4=10\,000$ would be $x=\pm\sqrt[4]{10\,000}=\pm10$ .

a) Since the power in $x^\var{intpower}=\var{intrhs}$ is even we will take the plus or minus $\var{intpower}$ nd rd th root to get two solutions.

$x^\var{intpower}$	$=$	$\var{intrhs}$

$\sqrt[\var{intpower}]{x^\var{intpower}}$	$=$	$\pm\sqrt[\var{intpower}]{\var{intrhs}}$

$x$	$=$	$\pm\var{intsoln}$

That is, $x$ equals $-\var{intsoln}$ or $\var{intsoln}$ .

b) Given $\simplify{{bxcoeff}y^{bpower}+{bb}}=\var{bc}$ , we can rearrange the equation to get $y^\var{bpower}$ by itself and then we can take the plus or minus $\var{bpower}$ nd root to get $y$ by itself.

$\simplify{{bxcoeff}y^{bpower}+{bb}}$	$=$	$\var{bc}$

$\simplify{{bxcoeff}y^{bpower}}$	$=$	$\simplify[basic]{{bc}-{bb}}$

$\simplify{{bxcoeff}y^{bpower}}$	$=$	$\simplify{{bc-bb}}$

$y^\var{bpower}$	$=$	$\simplify[!basic]{{bc-bb}/{bxcoeff}}$

$y^\var{bpower}$	$=$	$\simplify{{bc-bb}/{bxcoeff}}$

$\sqrt[\var{bpower}]{y^\var{bpower}}$	$=$	$\pm\sqrt[\var{bpower}]{\var{brhs}}$

$y$	$=$	$\pm\var{bsoln}$

That is, $y$ equals $-\var{bsoln}$ , or $\var{bsoln}$ .

c) Given $\simplify{{cxcoeff}z^{cpower}+{cb}}=\var{cc}$ , we can rearrange the equation to get $z^\var{cpower}$ by itself and then we can take the plus or minus $\var{cpower}$ nd root to get $z$ by itself.

$\simplify{{cxcoeff}z^{cpower}+{cb}}$	$=$	$\var{cc}$

$\simplify{{cxcoeff}z^{cpower}}$	$=$	$\simplify[basic]{{cc}-{cb}}$

$\simplify{{cxcoeff}z^{cpower}}$	$=$	$\simplify{{cc-cb}}$

$z^\var{cpower}$	$=$	$\simplify[!basic]{{cc-cb}/{cxcoeff}}$

$z^\var{cpower}$	$=$	$\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}$

$\sqrt[\var{cpower}]{z^\var{cpower}}$	$=$	$\pm\sqrt[\var{cpower}]{\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$

$z$	$=$	$\pm\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}^{1/{cpower}}}$

That is, $z$ equals $-\sqrt[\var{cpower}]{\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$ , or $\sqrt[\var{cpower}]{\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$ .

d) Given $\displaystyle{\simplify{((z+{db})^{bpower})/({ddenom})}}=\var{dc}$ , we can rearrange the equation to get $\simplify{(z+{db})^{bpower}}$ by itself, then we can take the plus or minus $\var{bpower}$ nd root of both sides to get $\simplify{z+{db}}$ by itself, and then rearrange to get $z$ by itself.

$\displaystyle{\simplify{((z+{db})^{bpower})/({ddenom})}}$	$=$	$\var{dc}$

$\simplify{(z+{db})^{bpower}}$	$=$	$\simplify[basic]{{dc}*{ddenom}}$

$\simplify{(z+{db})^{bpower}}$	$=$	$\var{dc*ddenom}$

$\sqrt[\var{bpower}]{\simplify{(z+{db})^{bpower}}}$	$=$	$\pm\sqrt[\var{bpower}]{\var{dc*ddenom}}$

$\simplify{z+{db}}$	$=$	$\pm\sqrt[\var{bpower}]{\var{dc*ddenom}}$

$z$	$=$	$\pm\simplify{root({dc*ddenom},{bpower})-{db}}$

$z$	$=$	$\pm\simplify{{dc*ddenom}^(1/{bpower})-{db}}$

That is, $z$ equals $-\simplify{root({dc*ddenom},{bpower})-{db}}$ , or $\simplify{root({dc*ddenom},{bpower})-{db}}$ .

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Solve equations which include a single even power (e.g. x^even=blah)

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

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Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Solve equations which include a single even power (e.g. x^even=blah)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`