Simplify the aritmetic expression: #[3/4 ·1/4 ·(5− 3/2)-: (3/4 − 3/16)] -: 7/4 ·(2 + 1/2)^2 −(1 + 1/2)^2#?

Answer 1

#23/12#

Given,

#[3/4*1/4*(5-3/2)-:(3/4-3/16)]-:7/4*(2+1/2)^2-(1+1/2)^2#

According to B.E.D.M.A.S., start by simplifying the round bracketed terms in the square brackets.

#=[3/4*1/4*(color(blue)(10/2)-3/2)-:(color(blue)(12/16)-3/16)]-:7/4*(2+1/2)^2-(1+1/2)^2#
#=[3/4*1/4*(color(blue)(7/2))-:(color(blue)(9/16))]-:7/4*(2+1/2)^2-(1+1/2)^2#

Omit the round brackets in the square brackets.

#=[3/4*1/4*7/2-:9/16]-:7/4*(2+1/2)^2-(1+1/2)^2#

Simplify the expression within the square brackets.

#=[3/16*7/2-:9/16]-:7/4*(2+1/2)^2-(1+1/2)^2#
#=[21/32*16/9]-:7/4*(2+1/2)^2-(1+1/2)^2#
#=[(21color(red)(-:3))/(32color(purple)(-:16)) * (16color(purple)(-:16))/(9color(red)(-:3))]-:7/4*(2+1/2)^2-(1+1/2)^2#
#=[7/2*1/3]-:7/4*(2+1/2)^2-(1+1/2)^2#
#=[7/6]-:7/4*(2+1/2)^2-(1+1/2)^2#

Omit the square brackets since the term is already simplified.

#=7/6-:7/4*(2+1/2)^2-(1+1/2)^2#

Continue simplifying the terms in the round brackets.

#=7/6-:7/4*(4/2+1/2)^2-(2/2+1/2)^2#
#=7/6-:7/4*(5/2)^2-(3/2)^2#
#=7/6-:7/4*(25/4)-(9/4)#

Omit the round brackets since the bracketed terms are already simplified.

#=7/6-:7/4*25/4-9/4#
#=7/6*4/7*25/4-9/4#
The #7#'s and #4#'s cancel each other out since they appear in the numerator and denominator as a pair.
#=color(red)cancelcolor(black)7/6*color(purple)cancelcolor(black)4/color(red)cancelcolor(black)7*25/color(purple)cancelcolor(black)4-9/4#
#=25/6-9/4#

Change the denominator of each fraction such that both fractions have the same denominator.

#=25/color(red)6(color(purple)4/color(purple)4)-9/color(purple)4(color(red)6/color(red)6)#
#=100/24-54/24#
#=46/24#
#=23/12#
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Answer 2

To simplify the arithmetic expression, let's break it down step by step:

  1. Start by simplifying the expressions within parentheses and brackets.
  2. Perform multiplications and divisions.
  3. Finally, compute the result.

[ \left[ \frac{3}{4} \cdot \frac{1}{4} \cdot \left( \sqrt{5} - \frac{3}{2} \right) - \frac{1}{3/4 - 3/16} \right] - \frac{7}{4} \cdot \left( 2 + \frac{1}{2} \right)^2 - \left( 1 + \frac{1}{2} \right)^2 ]

Let's simplify it step by step:

  1. Inside the brackets: [ \sqrt{5} - \frac{3}{2} = \frac{2\sqrt{5} - 3}{2} ] [ \frac{1}{3/4 - 3/16} = \frac{1}{3/16} = \frac{16}{3} ]

  2. Now, replace these values back into the expression: [ \left[ \frac{3}{4} \cdot \frac{1}{4} \cdot \left( \frac{2\sqrt{5} - 3}{2} \right) - \frac{16}{3} \right] - \frac{7}{4} \cdot \left( 2 + \frac{1}{2} \right)^2 - \left( 1 + \frac{1}{2} \right)^2 ]

  3. Compute the squares: [ \left( 2 + \frac{1}{2} \right)^2 = \left( \frac{5}{2} \right)^2 = \frac{25}{4} ] [ \left( 1 + \frac{1}{2} \right)^2 = \left( \frac{3}{2} \right)^2 = \frac{9}{4} ]

  4. Now, perform the multiplications: [ \frac{3}{4} \cdot \frac{1}{4} \cdot \frac{2\sqrt{5} - 3}{2} = \frac{3}{32} \cdot (2\sqrt{5} - 3) = \frac{3\sqrt{5}}{16} - \frac{9}{32} ]

  5. Now, replace these values back into the expression: [ \left( \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{16}{3} \right) - \frac{7}{4} \cdot \frac{25}{4} - \frac{9}{4} ]

  6. Combine like terms and perform the remaining arithmetic operations.

[ \left( \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{16}{3} \right) - \frac{7 \cdot 25}{4 \cdot 4} - \frac{9}{4} ]

[ = \left( \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{64}{3} \right) - \frac{175}{16} - \frac{9}{4} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{64}{3} - \frac{175}{16} - \frac{9}{4} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{256}{32} - \frac{420}{16} - \frac{36}{16} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{256}{32} - \frac{420}{16} - \frac{36}{16} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{256 + 420 + 36}{16} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{712}{16} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{356}{8} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{178}{4} ]

[ = \frac{3\sqrt{5}}{16} - \frac{9}{32} - \frac{89}{2} ]

[ \approx \frac{3\sqrt{5}}{16} - 0.28125 - 44.5 ]

[ \approx \frac{3\sqrt{5}}{16} - 44.78125 ]

[ \approx -44.78125 + \frac{3\sqrt{5}}{16} ]

[ \approx \frac{3\sqrt{5}}{16} - 44.78125 ]

So, the simplified arithmetic expression is approximately ( \frac{3\sqrt{5}}{16} - 44.78125 ).

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Answer 3

The simplified arithmetic expression is ( -\frac{33}{4} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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