Rearranging Equations – Dealing with Fractions Reinforcement Questions

1.

Isolate b in the following equation:

begin mathsize 14px style fraction numerator left parenthesis R subscript x right parenthesis to the power of 6 over denominator n proportional to to the power of 4 end fraction space equals space b over element of space fraction numerator L o g subscript 10 increment T over denominator infinity C D to the power of empty set end fraction end style

First, we move begin mathsize 14px style fraction numerator L o g subscript 10 increment T over denominator infinity C D to the power of empty set end fraction end style across the equals sign by multiplying both sides by begin mathsize 14px style fraction numerator infinity C D to the power of empty set over denominator L o g subscript 10 space increment T end subscript end fraction end style :

begin mathsize 14px style fraction numerator infinity C D to the power of empty set over denominator L o g subscript 10 space increment T end subscript end fraction space fraction numerator left parenthesis R subscript x right parenthesis to the power of 6 over denominator n proportional to to the power of 4 end fraction space equals space b over element of end style

And then all we need to do is multiply both sides by element of:

begin mathsize 14px style b space equals space element of space fraction numerator infinity C D to the power of empty set over denominator L o g subscript 10 space increment straight T end subscript space space end fraction space fraction numerator left parenthesis R subscript x right parenthesis to the power of 6 over denominator n proportional to to the power of 4 end fraction space end style

2.

Rearrange this equation to isolate begin mathsize 14px style empty set end style:

begin mathsize 14px style fraction numerator begin display style fraction numerator c b to the power of r over denominator empty set end fraction end style over denominator begin display style fraction numerator begin display style x subscript 1 space minus space x subscript 2 end style over denominator increment delta to the power of 4 end fraction end style end fraction space equals space A space plus space R T end style

First, we can movebegin mathsize 14px style fraction numerator begin display style x subscript 1 space minus space x subscript 2 end style over denominator increment delta to the power of 4 end fraction end style over across by multiplying both sides by it:

begin mathsize 14px style fraction numerator c b to the power of r over denominator empty set end fraction space equals space left parenthesis A space plus space R T right parenthesis space fraction numerator begin display style x subscript 1 space minus space x subscript 2 end style over denominator increment delta to the power of 4 end fraction end style

We can’t move the numerator over, so we next have to move the begin mathsize 14px style fraction numerator begin display style x subscript 1 space end subscript minus space x subscript 2 end style over denominator increment delta to the power of 4 end fraction end style back by multiplying both sides by its reciprocal begin mathsize 14px style fraction numerator begin display style increment delta to the power of 4 end style over denominator x subscript 1 space minus space x subscript 2 end fraction end style:

begin mathsize 14px style fraction numerator c b to the power of r over denominator empty set end fraction space fraction numerator begin display style increment delta to the power of 4 end style over denominator x subscript 1 space minus space x subscript 2 end fraction space equals space left parenthesis A space plus space R T right parenthesis end style

Now we can move begin mathsize 14px style fraction numerator c b to the power of r over denominator empty set end fraction end style across the equals sign by multiplying both sides by its reciprocal begin mathsize 14px style fraction numerator theta over denominator c b to the power of r end fraction end style :

begin mathsize 14px style fraction numerator begin display style increment delta to the power of 4 end style over denominator x subscript 1 space minus space x subscript 2 end fraction space equals space left parenthesis A space plus space R T right parenthesis space fraction numerator theta over denominator c b to the power of r end fraction end style

And we can move begin mathsize 14px style c b to the power of r end style  back over by multiplying by both sides by it:

begin mathsize 14px style open parentheses fraction numerator begin display style increment delta to the power of 4 end style over denominator x subscript 1 space minus space x subscript 2 end fraction close parentheses space c b to the power of r space equals space left parenthesis A space plus space R T right parenthesis space theta end style

And finally, we can move begin mathsize 14px style left parenthesis space A space plus space R T space right parenthesis end style across by dividing both sides by it: 

begin mathsize 14px style fraction numerator open parentheses fraction numerator begin display style increment delta to the power of 4 end style over denominator x subscript 1 space minus space x subscript 2 end fraction close parentheses over denominator space left parenthesis A space plus space R T right parenthesis end fraction space c b to the power of r space equals space theta end style

3.

Isolate G in the following equation:

begin mathsize 14px style F space fraction numerator b G over denominator n R T end fraction space equals space fraction numerator space delta to the power of left parenthesis t s subscript 1 space end subscript minus space t s subscript 2 right parenthesis end exponent over denominator W H subscript b end fraction space plus space fraction numerator C U to the power of intersection over denominator t to the power of 9 end fraction end style

First, we can move the F by dividing both sides by it:

begin mathsize 16px style fraction numerator b G over denominator n R T end fraction space equals space fraction numerator begin display style fraction numerator delta to the power of left parenthesis t s subscript 1 space minus space t s subscript 2 right parenthesis end exponent over denominator W H subscript b end fraction end style space plus space begin display style fraction numerator C U to the power of intersection over denominator t to the power of 9 end fraction end style over denominator F end fraction end style

Now we can move begin mathsize 14px style n R T end style by multiplying both sides by it:

begin mathsize 16px style b G space equals space open parentheses fraction numerator begin display style fraction numerator delta to the power of left parenthesis t s subscript 1 space minus space t s subscript 2 right parenthesis end exponent over denominator W H subscript b end fraction space plus space fraction numerator C U to the power of intersection over denominator t to the power of 9 end fraction end style over denominator F end fraction close parentheses space n R T end style

And finally, move the b by dividing both sides by it:

begin mathsize 16px style G space equals space fraction numerator open parentheses fraction numerator begin display style fraction numerator delta to the power of left parenthesis t s subscript 1 space minus space t s subscript 2 right parenthesis end exponent over denominator W H subscript b end fraction space plus space fraction numerator C U to the power of intersection over denominator t to the power of 9 end fraction end style over denominator F end fraction close parentheses space n R T over denominator b end fraction end style

4.

Isolate F in this equation:

begin mathsize 14px style fraction numerator begin display style fraction numerator begin display style 1 half space end style F over denominator begin display style x over y end style end fraction end style over denominator begin display style fraction numerator b space cross times space 10 to the power of negative 12 end exponent over denominator begin display style 1 half end style R T end fraction end style end fraction space equals space D squared end style

Start by moving begin mathsize 14px style fraction numerator b space cross times space 10 to the power of negative 12 end exponent over denominator begin display style 1 half end style R T end fraction end style by multiplying both sides by its reciprocal begin mathsize 14px style fraction numerator 1 half space R T over denominator b space cross times space 10 to the power of negative 12 end exponent end fraction end style :

begin mathsize 14px style fraction numerator begin display style 1 half end style F over denominator begin display style x over y end style end fraction space equals space D squared space open parentheses fraction numerator b space cross times space 10 to the power of negative 12 end exponent over denominator begin display style 1 half R T end style end fraction close parentheses end style

Now we can move begin mathsize 14px style x over y end style by multiplying both sides by it:

begin mathsize 14px style 1 half space F space equals space D to the power of 2 space end exponent space open parentheses fraction numerator b space cross times space 10 to the power of negative 12 end exponent over denominator begin display style 1 half R T end style end fraction close parentheses space space x over y end style

Finally, we can move the begin mathsize 14px style 1 half end style across by multiplying both sides by its reciprocal begin mathsize 14px style 2 over 1 end style. Any number with 1 as the denominator has the same value as the numerator, so we multiply the right-hand side by 2:

begin mathsize 14px style F space equals space 2 D squared space space open parentheses fraction numerator b space cross times space 10 to the power of negative 12 end exponent over denominator begin display style 1 half end style R T end fraction close parentheses space x over y end style

5.

Rearrange the equation below to isolate C:

begin mathsize 14px style fraction numerator C to the power of 2 space end exponent over denominator X subscript 1 to the power of 5 space plus space R beta rho end fraction fraction numerator begin display style 1 half end style space v over denominator F to the power of 4 to the power of a end fraction space equals space 25 space left parenthesis S r to the power of 4 space plus space 3 over 4 space b right parenthesis end style

First, we can move begin mathsize 14px style fraction numerator begin display style 1 half end style v over denominator F to the power of 4 to the power of a end exponent end fraction end style across the equals sign by multiplying both sides by its reciprocal begin mathsize 14px style fraction numerator begin display style F to the power of 4 to the power of a end exponent end style over denominator begin display style 1 half v end style end fraction end style:

begin mathsize 14px style fraction numerator C to the power of 2 space end exponent over denominator X subscript 1 to the power of 5 space plus space R beta rho end fraction space equals space 25 space open parentheses S r to the power of 4 space plus space 3 over 4 b close parentheses space open parentheses fraction numerator begin display style F to the power of 4 to the power of a end style over denominator begin display style 1 half end style v end fraction close parentheses end style

Next, we move begin mathsize 14px style X subscript 1 to the power of 5 space plus space R beta rho end style across by multiplying both sides by it:

begin mathsize 14px style C to the power of 2 space end exponent equals space 25 space open parentheses S r to the power of 4 space plus space 3 over 4 b close parentheses space space open parentheses fraction numerator begin display style F to the power of 4 to the power of a end style over denominator begin display style 1 half end style v end fraction close parentheses space space left parenthesis X subscript 1 to the power of 5 space plus space R beta rho right parenthesis end style

Finally, we take the square root of both sides to isolate C:

begin mathsize 14px style C space equals space square root of 25 space open parentheses S r to the power of 4 space plus space 3 over 4 b close parentheses space open parentheses fraction numerator F to the power of 4 to the power of a end exponent over denominator begin display style 1 half end style v end fraction close parentheses space left parenthesis X subscript 1 to the power of 5 space plus space R beta rho right parenthesis end root end style

We expand our view of equations by encompassing brackets, next.

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