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3 4 5 8 = 3 4 ÷ 5 8

Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

Simplify complex fractions.

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator.
  4. Simplify if possible.

Simplify: ( 1 2 ) 2 4 + 3 2 .

Solution

( 1 2 ) 2 4 + 3 2
Simplify the numerator. 1 4 4 + 3 2
Simplify the term with the exponent in the denominator. 1 4 4 + 9
Add the terms in the denominator. 1 4 13
Divide the numerator by the denominator. 1 4 ÷ 13
Rewrite as multiplication by the reciprocal. 1 4 · 1 13
Multiply. 1 52
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Simplify: ( 1 3 ) 2 2 3 + 2 .

1 90

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Simplify: 1 + 4 2 ( 1 4 ) 2 .

272

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Simplify: 1 2 + 2 3 3 4 1 6 .

Solution

1 2 + 2 3 3 4 1 6
Rewrite numerator with the LCD of 6 and denominator with LCD of 12. 3 6 + 4 6 9 12 2 12
Add in the numerator. Subtract in the denominator. 7 6 7 12
Divide the numerator by the denominator. 7 6 ÷ 7 12
Rewrite as multiplication by the reciprocal. 7 6 · 12 7
Rewrite, showing common factors. 7 · 6 · 2 6 · 7 · 1
Simplify. 2
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Simplify: 1 3 + 1 2 3 4 1 3 .

2

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Simplify: 2 3 1 2 1 4 + 1 3 .

2 7

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Evaluate variable expressions with fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Evaluate x + 1 3 when

  1. x = 1 3
  2. x = 3 4 .

Solution

To evaluate x + 1 3 when x = 1 3 , substitute 1 3 for x in the expression.

x + 1 3
. .
Simplify. 0

To evaluate x + 1 3 when x = 3 4 , we substitute 3 4 for x in the expression.

x + 1 3
. .
Rewrite as equivalent fractions with the LCD, 12. 3 · 3 4 · 3 + 1 · 4 3 · 4
Simplify the numerators and denominators. 9 12 + 4 12
Add. 5 12
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Evaluate: x + 3 4 when

  1. x = 7 4
  2. x = 5 4

  1. −1
  2. 1 2

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Evaluate: y + 1 2 when

  1. y = 2 3
  2. y = 3 4

  1. 7 6
  2. 1 4

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Evaluate y 5 6 when y = 2 3 .

Solution

We substitute 2 3 for y in the expression.

y 5 6
. .
Rewrite as equivalent fractions with the LCD, 6. 4 6 5 6
Subtract. 9 6
Simplify. 3 2
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Evaluate: y 1 2 when y = 1 4 .

3 4

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Evaluate: x 3 8 when x = 5 2 .

23 8

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Evaluate 2 x 2 y when x = 1 4 and y = 2 3 .

Solution

Substitute the values into the expression. In 2 x 2 y , the exponent applies only to x .

.
. .
Simplify exponents first. .
Multiply. The product will be negative. .
Simplify. .
Remove the common factors. .
Simplify. .
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Evaluate. 3 a b 2 when a = 2 3 and b = 1 2 .

1 2

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Evaluate. 4 c 3 d when c = 1 2 and d = 4 3 .

2 3

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Evaluate p + q r when p = −4 , q = −2 , and r = 8 .

Solution

We substitute the values into the expression and simplify.

p + q r
. .
Add in the numerator first. 6 8
Simplify. 3 4
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Evaluate: a + b c when a = −8 , b = −7 , and c = 6 .

5 2

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Evaluate: x + y z when x = 9 , y = −18 , and z = −6 .

3 2

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Key concepts

  • Find the least common denominator (LCD) of two fractions.
    1. Factor each denominator into its primes.
    2. List the primes, matching primes in columns when possible.
    3. Bring down the columns.
    4. Multiply the factors. The product is the LCM of the denominators.
    5. The LCM of the denominators is the LCD of the fractions.
  • Equivalent Fractions Property
    • If a , b , and c are whole numbers where b 0 , c 0 then
      a b = a c b c and a c b c = a b
  • Convert two fractions to equivalent fractions with their LCD as the common denominator.
    1. Find the LCD.
    2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
    3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
    4. Simplify the numerator and denominator.
  • Add or subtract fractions with different denominators.
    1. Find the LCD.
    2. Convert each fraction to an equivalent form with the LCD as the denominator.
    3. Add or subtract the fractions.
    4. Write the result in simplified form.
  • Summary of Fraction Operations
    • Fraction multiplication: Multiply the numerators and multiply the denominators.
      a b c d = a c b d
    • Fraction division: Multiply the first fraction by the reciprocal of the second.
      a b + c d = a b d c
    • Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
      a c + b c = a + b c
    • Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
      a c - b c = a - b c
  • Simplify complex fractions.
    1. Simplify the numerator.
    2. Simplify the denominator.
    3. Divide the numerator by the denominator.
    4. Simplify if possible.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
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yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Practice Key Terms 1

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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