Question

Last question thanks guys!! (only 1!)

Answer 1

The correct answer is:  [C]:  " x² = √32 " .
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Explanation:
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Note:  Working backward,   "(± 4√2)" , squared, equals what value(s)?
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Note:  We are actually given 2 (TWO) solutions:

  " +4√2" and "–4√2" ; 
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√32 = √8 √4 = √4 √2 √4 = 4√2 ;

- √32 = -1 * √32  =   - 4√2
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or:  

√32 = √16 √2  = 4√2 ; 

-1 * √32  =   - 4√2
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However, BOTH these values, when "squared" (i.e. raised to the exponential power of "two"; will result in the same value— which is:  "32" ; 
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 is equal to:  "(4√2)²  = (4)² *(√2)²  =  (4*4) (√2*√2) = 16*2 = " 32 " .

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√32 = √8 √4 = √4 √2 √4 = 4√2 ;

- √32 = -1 * √32  =   - 4√2
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or:  

√32 = √16 √2  = 4√2 ; 

-1 * √32  =   - 4√2
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          " (4√2)²  =  (4)²  * (√2)²  =  (4 * 4) (√2*√2)  =  16 * 2 = " 32 " .  

         " (-4√2)²  = (-4)²  * (√2)²  = (-4 *-4) (2) = 16 * 2 = " 32 " .
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The correct answer is:  " 32 " ; which is:  Answer choice:  [D]:  " x² = 32 " .

We know that the answer is: " " {" ± 8 "}; since we are dealing with equations that contain "x SQUARED"; that is, " x²"; and when we solve for the 'square root of all values of a [variable raised to an even positive integer] ;

we take the "plus or minus" square root values ;  since the value, "x" could be plus or minus;  since a "negative value" multiplied by a "negative value" is a "positive value" ; 
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So, the correct answer is:  Answer choice:  [B]:  " x² = ± 8 " .
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Let us check our answer:  

√32 = √16*√2 = 4√2 ;

– √32  =  – 4√2 ;
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Also, note:
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   x²  =  32 ;  Solve for "x" ; 

→ Take the square root of "each side" of the equation ; 
     to isolate "x" on one side of the equation; & to solve for "x" ;

→  √(x²)  =  √32 ; 

→  | x |   = √32 ;

→  x = ±   √32 . Yes!
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Answer 2

$x=\pm4 \sqrt{2} \\ x^2=(4 \sqrt{2} )^2 \\ x^2=4^2\cdot(\sqrt{2} )^2 \\ x^2=16\cdot (2) \\ x^2=32$