- #1
sambarbarian
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Find the value of [itex] \sqrt{-\sqrt{3}+\sqrt{3 + 8 \sqrt{7 + 4\sqrt{3}}}} [/itex]the options are [tex]1[/tex] , [tex]0[/tex] , [tex]2[/tex] , [tex]3[/tex]
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sambarbarian said:oh , sorry again , the options are 1 , 0 , 2 and 3
Mentallic said:Well just work backwards from the most inner radical and take approximations.
For the answer to be equal to 0, we need to have
[tex]\sqrt{-\sqrt{3}+\sqrt{3}}[/tex]
which we clearly don't. For 1 we need
[tex]\sqrt{-\sqrt{3}+(1+\sqrt{3})}[/tex]
And using the approximation of [itex]\sqrt{3}\approx1.7[/itex] would suffice.
For 2 we need
[tex]\sqrt{-\sqrt{3}+(4+\sqrt{3})}[/tex]
And finally for 3 we need
[tex]\sqrt{-\sqrt{3}+(9+\sqrt{3})}[/tex]
So what is the radical
[tex]\sqrt{3+8\sqrt{7+4\sqrt{3}}}[/tex] closest to? 2.7, 5.7 or 10.7?
Ray Vickson said:Although I find it hard to believe, the original expression actually does come out exactly to a small integer value.
RGV
Pranav-Arora said:Start by writing 7+4√3 as 4+4√3+3 which can be simplified to (2+√3)^2.
Pranav-Arora said:Start by writing 7+4√3 as 4+4√3+3 which can be simplified to (2+√3)^2.
Yes, Pranav-Arora !ehild said:Ingenious Pranav! And the same method can be applied again to get a small integer as result.
ehild
An exponent is a number that represents how many times a base number should be multiplied by itself. It is typically written as a superscript number to the right of the base number.
To calculate the value of an exponent, you can use the formula an = a x a x a... (n times), where a is the base number and n is the exponent. For example, 23 = 2 x 2 x 2 = 8.
A positive exponent indicates that the base number should be multiplied by itself a certain number of times. A negative exponent, on the other hand, indicates that the base number should be divided by itself a certain number of times. For example, 23 = 2 x 2 x 2 = 8, while 2-3 = 1/(2 x 2 x 2) = 1/8.
If the bases are the same, you can simply add or subtract the exponents. For example, 23 x 24 = 27. If the bases are different, you can use the property (am)n = am x n. For example, (23)4 = 23 x 4 = 212.
Yes, exponents can be fractions or decimals. For example, 21/2 = √2 and 20.5 = √2. This is known as a radical or fractional exponent. It represents the root of the base number. In general, a1/n = √na.