Answer :

To solve these cube root expressions, we'll simplify each one step-by-step. Cube roots require us to find a number that, when multiplied by itself three times, gives us the original number.

Let's break down each expression:

  1. [tex]\sqrt[3]{8x^6 + 19x^6}[/tex]

    First, simplify inside the cube root:
    [tex]8x^6 + 19x^6 = 27x^6[/tex]

    Now, find the cube root:
    [tex]\sqrt[3]{27x^6} = \sqrt[3]{27} \times \sqrt[3]{x^6}[/tex]

    Since [tex]27 = 3^3[/tex], we have:
    [tex]\sqrt[3]{27} = 3[/tex]

    For [tex]x^6[/tex], [tex]\sqrt[3]{x^6} = x^2[/tex] because [tex](x^2)^3 = x^6[/tex].

    So, the whole expression simplifies to:
    [tex]3x^2[/tex]

  2. [tex]\sqrt[3]{25m^9 + 100m^9}[/tex]

    Simplify inside the cube root:
    [tex]25m^9 + 100m^9 = 125m^9[/tex]

    Now, find the cube root:
    [tex]\sqrt[3]{125m^9} = \sqrt[3]{125} \times \sqrt[3]{m^9}[/tex]

    Since [tex]125 = 5^3[/tex], we have:
    [tex]\sqrt[3]{125} = 5[/tex]

    For [tex]m^9[/tex], [tex]\sqrt[3]{m^9} = m^3[/tex] because [tex](m^3)^3 = m^9[/tex].

    So, the simplified expression is:
    [tex]5m^3[/tex]

  3. [tex]\sqrt[3]{(4x^3 + 4x^3)(2x^3)}[/tex]

    Simplify inside the brackets first:
    [tex]4x^3 + 4x^3 = 8x^3[/tex]

    Now multiply by [tex]2x^3[/tex]:
    [tex](8x^3)(2x^3) = 16x^6[/tex]

    Find the cube root of the result:
    [tex]\sqrt[3]{16x^6} = \sqrt[3]{16} \times \sqrt[3]{x^6}[/tex]

    Since [tex]16 = 2^4 = (2^2)^2[/tex], we don't get a neat cube root for 16, so:
    [tex]\sqrt[3]{16} = 2^{4/3}[/tex] (usually left in this form unless more approximation is needed)

    For [tex]x^6[/tex], as before, [tex]\sqrt[3]{x^6} = x^2[/tex].

    Thus, the expression simplifies to:
    [tex]2^{4/3} x^2[/tex]

These steps demonstrate how to simplify cube roots by breaking down and working through each part of the expression carefully.

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