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Simplify the following expressions involving exponents: 1) 20100 ext / 20100, 2) racx32x3, 3) (-2xyz)0, 4) rac(2a5b3)3 (2a2b4)232a19b17

Mathematics

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Here are the solutions to the problems.

1) Simplify 20100÷2010020^{100} \div 20^{100}.

  1. Apply the division rule for exponents, am÷an=am−na^m \div a^n = a^{m-n}.

    20100÷20100=20100−10020^{100} \div 20^{100} = 20^{100-100}
  2. Simplify the exponent.

    20100−100=20020^{100-100} = 20^0
  3. Apply the rule a0=1a^0 = 1 for any non-zero base aa.

    200=120^0 = 1

    The simplified expression is 1\boxed{1}.

    2) Simplify x32x3\frac{x^3}{2x^3}.

  4. Separate the coefficient and variable terms.

    x32x3=12×x3x3\frac{x^3}{2x^3} = \frac{1}{2} \times \frac{x^3}{x^3}
  5. Simplify the variable term using the rule am÷am=1a^m \div a^m = 1.

    x3x3=x3−3=x0=1\frac{x^3}{x^3} = x^{3-3} = x^0 = 1
  6. Multiply the simplified parts.

    12×1=12\frac{1}{2} \times 1 = \frac{1}{2}

    The simplified expression is 12\boxed{\frac{1}{2}}.

    3) Simplify (−2xyz)0(-2xyz)^0.

  7. Apply the rule a0=1a^0 = 1 for any non-zero base aa.

    Since the entire expression (−2xyz)(-2xyz) is raised to the power of 0, the result is 1.

    (−2xyz)0=1(-2xyz)^0 = 1

    The simplified expression is 1\boxed{1}.

    4) Simplify (2a5b3)3(2a2b4)232a19b17\frac{(2a^5b^3)^3 (2a^2b^4)^2}{32a^{19}b^{17}}.

  8. Simplify the first term in the numerator, (2a5b3)3(2a^5b^3)^3.

    Apply (xy)n=xnyn(xy)^n = x^n y^n and (xm)n=xmn(x^m)^n = x^{mn}.

    (2a5b3)3=23(a5)3(b3)3=8a15b9(2a^5b^3)^3 = 2^3 (a^5)^3 (b^3)^3 = 8a^{15}b^9
  9. Simplify the second term in the numerator, (2a2b4)2(2a^2b^4)^2.

    Apply (xy)n=xnyn(xy)^n = x^n y^n and (xm)n=xmn(x^m)^n = x^{mn}.

    (2a2b4)2=22(a2)2(b4)2=4a4b8(2a^2b^4)^2 = 2^2 (a^2)^2 (b^4)^2 = 4a^4b^8
  10. Multiply the simplified terms in the numerator.

    (8a15b9)(4a4b8)=(8×4)(a15a4)(b9b8)(8a^{15}b^9)(4a^4b^8) = (8 \times 4)(a^{15}a^4)(b^9b^8)

    Apply aman=am+na^m a^n = a^{m+n}.

    32a15+4b9+8=32a19b1732a^{15+4}b^{9+8} = 32a^{19}b^{17}
  11. Substitute the simplified numerator back into the original expression.

    32a19b1732a19b17\frac{32a^{19}b^{17}}{32a^{19}b^{17}}
  12. Simplify the entire fraction.

    Since the numerator and denominator are identical, the expression simplifies to 1.

    32a19b1732a19b17=1\frac{32a^{19}b^{17}}{32a^{19}b^{17}} = 1

    The simplified expression is 1\boxed{1}.

    5) Simplify (212a2vfq2w2ercsd9f10q3)0\left(\frac{2^{12}a^2vfq^2w^2er}{csd^9f^{10}q^3}\right)^0.

  13. Apply the rule a0=1a^0 = 1 for any non-zero base aa.

    Since the entire fraction inside the parentheses is raised to the power of 0, the result is 1.

    (212a2vfq2w2ercsd9f10q3)0=1\left(\frac{2^{12}a^2vfq^2w^2er}{csd^9f^{10}q^3}\right)^0 = 1

    The simplified expression is 1\boxed{1}.

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