Math Problem Statement

Rewrite the expressions without zero or negative exponents.

Solution

Here’s the solution for the given problems, rewriting each expression without zero or negative exponents:

  1. a3=1a3a^{-3} = \frac{1}{a^3}

  2. 3x0=33x^0 = 3 (since x0=1x^0 = 1)

  3. m5n0=m5m^5n^0 = m^5 (since n0=1n^0 = 1)

  4. (4a2)0=1(4a^2)^0 = 1 (any nonzero number to the power of 0 equals 1)

  5. 2m1=2m2m^{-1} = \frac{2}{m}

  6. (xy)3=1(xy)3(xy)^{-3} = \frac{1}{(xy)^3}

  7. (2a)3=1(2a)3=18a3(2a)^{-3} = \frac{1}{(2a)^3} = \frac{1}{8a^3}

  8. 5y2=5y2-5y^{-2} = -\frac{5}{y^2}

  9. a3b=1a3b\frac{a^{-3}}{b} = \frac{1}{a^3b}

  10. x5y3=y3x5\frac{x^{-5}}{y^{-3}} = \frac{y^3}{x^5}

  11. (8ab)0=1-(8ab)^0 = -1 (since (8ab)0=1(8ab)^0 = 1)

  12. a1(bc)0=1aa^{-1}(bc)^0 = \frac{1}{a} (since (bc)0=1(bc)^0 = 1)

  13. 3y4=3y43y^{-4} = \frac{3}{y^4}

  14. (m2n0)0p2=1p2(m^2n^0)^0p^{-2} = \frac{1}{p^2} (since (m2n0)0=1(m^2n^0)^0 = 1)

  15. a2b3=1a21b3a^{-2} - b^{-3} = \frac{1}{a^2} - \frac{1}{b^3}

  16. x4b3x^4b^3 (already simplified)

  17. (15a6b3c8)0=1(15a^6b^3c^8)^0 = 1

  18. (12x5y3z7)1=112x5y3z7(12x^5y^3z^7)^{-1} = \frac{1}{12x^5y^3z^7}

  19. x25=15x2\frac{x^{-2}}{5} = \frac{1}{5x^2}

  20. (2m53n4)0=1\left( \frac{2m^{-5}}{3n^{-4}} \right)^0 = 1 (anything to the power of 0 equals 1)

Would you like further details or explanations for any of these steps?


Five follow-up questions:

  1. Why does any nonzero number raised to the power of 0 equal 1?
  2. How do negative exponents change an expression?
  3. What are some real-world applications of exponents in mathematics?
  4. How does fractional exponents differ from whole-number exponents?
  5. Can you explain the difference between an exponent of 0 and a negative exponent?

Tip:

Always simplify expressions step by step, handling parentheses, exponents, and negative signs systematically for clarity.

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Math Problem Analysis

Mathematical Concepts

Exponents
Negative Exponents
Simplification of Algebraic Expressions

Formulas

a^(-n) = 1 / a^n (for non-zero a)
(a^m)(a^n) = a^(m+n)
(a^m)^n = a^(mn)

Theorems

Zero Exponent Rule: a^0 = 1 (for non-zero a)
Negative Exponent Rule

Suitable Grade Level

Grades 8-10