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# Which of the following is closest to the value of (√21 - √7)^2?

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Which of the following is closest to the value of (√21 - √7)^2?  [#permalink]

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30 Jun 2018, 23:53
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Which of the following is closest to the value of $$(\sqrt{21}-\sqrt{7})^2$$?

A. 4
B. 5
C. 7
D. 16
E. 196

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Re: Which of the following is closest to the value of (√21 - √7)^2?  [#permalink]

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01 Jul 2018, 00:07
this is of the form a2-2ab+b2
so we have (sqrt21)^2 - 2 sqrt(21*7) + sqrt7^2
= 21 - 2(sqrt 147) + 7
=28- 2 (sumthin more little more than 12) as 12 sqare is 144
=28-24 = 4 , now ans should be littler less than 4 as we have srt 147 , which is little more than 12

so ans 3
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Which of the following is closest to the value of (√21 - √7)^2?  [#permalink]

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01 Jul 2018, 02:43
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Bunuel wrote:
Which of the following is closest to the value of $$(\sqrt{21}-\sqrt{7})^2$$?

A. 4
B. 5
C. 7
D. 16
E. 196

21 lies between $$16(4^2)$$ and $$25(5^2)$$ and the square root of 21 is little over 4.5 $$(4.5^2 = 20.25)$$
Similarly, 7 lies between $$4(2^2)$$ and $$9(3^2)$$ and the square root of 7 is a little over 2.6 $$(2.6^2 = 6.76)$$

Therefore, the value closest to $$(\sqrt{21}-\sqrt{7})^2$$ is $$(4.5 - 2.6)^2 = (1.9)^2$$ = approximately 4(Option A)
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Re: Which of the following is closest to the value of (√21 - √7)^2?  [#permalink]

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01 Jul 2018, 04:23
= 21+ 7- 2*sqrt21* sqrt 7
=28 - 2* sqrt(3*7*7)
=28 - 2* 7*sqrt3
=28 - 14*sqrt3
=28 - (14* 1.73)
=28- 24.24 = 3.76 ~ 4
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Which of the following is closest to the value of (√21 - √7)^2?  [#permalink]

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01 Jul 2018, 05:22
Bunuel wrote:
Which of the following is closest to the value of $$(\sqrt{21}-\sqrt{7})^2$$?

A. 4
B. 5
C. 7
D. 16
E. 196

Of course GMAT won't require to know square roots of anything more than 5 apart from perfect squares

$$(\sqrt{21}-\sqrt{7})^2=√21^2+√7^2-2*√21*√7=21+7-2*√3*√7*√7=28-14√3=28-14*1.73=14(2-1.73)=14*0.27$$
So slightly less than 14*0.3=4.2
So closest is 4

A
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Re: Which of the following is closest to the value of (√21 - √7)^2?  [#permalink]

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04 Jul 2018, 19:08
Bunuel wrote:
Which of the following is closest to the value of $$(\sqrt{21}-\sqrt{7})^2$$?

A. 4
B. 5
C. 7
D. 16
E. 196

Foiling we have:

(√21)^2 + (√7)^2 - 2(√21)(√7) = 21 + 7 - 2√147

Since √147 is a little more than 12, we have:

21 + 7 - 2(12) = 28 - 24 = 4

Alternate Solution:

We can use approximation to solve this problem quickly. We see that √21 is close to 5, and √7 is close to 3. Therefore (5 - 3)^2 = 4.

Alternate Solution:

We note that √21 = √(7*3) = (√7)*(√3). Then,

(√21 - √7)^2 = (√7*√3 - √7)^2 = [(√7)^2][√3 - 1]^2 = 7*(√3 - 1)^2

We note that √3 is approximately 1.7; thus we can approximate as

(√21 - √7)^2 ≈ 7 * (1.7 - 1)^2

(√21 - √7)^2 ≈ 7 * (0.7)^2

(√21 - √7)^2 ≈ 7 * (0.49) = 3.43

The closest value among the choices is A.

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Re: Which of the following is closest to the value of (√21 - √7)^2?  [#permalink]

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10 Mar 2019, 14:14
Here's the calculation if you estimate square roots:

(√3*√7 - √7)² = (1.7*2.65 - 2.65)² ...estimate 2.65 to 3 here... (3*[1.7 - 1])² = (3*.7)² = 2.1² ≈ 4, A
Re: Which of the following is closest to the value of (√21 - √7)^2?   [#permalink] 10 Mar 2019, 14:14
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