Why is 4√-16 not a real number, but 5√-32 is?

I need math help I thought a negative square root number is not a real number, but when I checked in the back of the book it said 5√-32= -2


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Most Helpful Girl

  • thats just life man.

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Most Helpful Guy

  • A negative mumber cannot have an even numbered root. Remember that neg * neg is always positive. So -2^2 =4 ( that is -2*-2=4) and -2^4=16 (-2*-2*-2*-2=16)
    So no matter how hard you try you cannot multiply a negative number by itself an even number of times and end up with a negative number. An odd root works nicely since neg*neg*neg will always be negative. So the 5th root of a negative number is a real number. But because it can't be calculated, the 4th root of a negative number is imaginary, not real.
    Hope this helps.

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What Girls Said 1

  • Ohhhhh I thought you meant 4 times (root of -16). Never mind.

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What Guys Said 1

  • The square roots of negative numbers are ALWAYS imaginary.

    That's square roots only. Don't forget to evaluate the whole expression.

    Maybe you're reading the book wrong? 5 times the square root of negative 32 is definitely not negative 2.

    So if that's what the book said, that's definitely a typo.

    5* sqrt (-32) = undefined, because negatives don't have square roots (in the traditional sense)

    In imaginaries it'd be i * sqrt (+32). sqrt (32) is gonna be some irrational number, 4*sqrt (2)

    so 5* sqrt (-32) = 20*i*sqrt (2)

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    • yeah that's what my book says :(

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    • So why is that though? @justphi

    • @Justiphi Oooh! Good catch! I knew it was the kid failing to parse.

      Okay. (-2) times itself 5 times is what? -2,+4,-8,+16,-32 Tada!

      The way to transcribe those tricky roots and powers in computer text is like this.
      ^
      That carrot is the power symbol. 4*4 can be written 4^2. that's four squared.
      (-2)^5 that's negative two to the 5th power.

      To express roots, you do the reciprocal. sqrt (2) is the same as 4^(1/2) = 2

      So you could have written it (-32)^(1/5)= -2

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