Exponents Worksheet

Simplify. Remember, any exponents in the final answer must be positive. The rules used to justify each step of the answer are in parentheses.

1) 21x²y/-15y⁴ = 7x²/-5y³

• Break down the exercise into pieces. Match constants with constants and variables with variables.
  21/-15 * x² * y/y⁴

• Whenever a variable doesn't have an exponent, automatically place a 1 in place of the exponent.
  21/-15 * x² * y¹/y⁴

• Simplify each piece.
  21/-15 = 7/-5
  x² = x² (Reflexive Property)
  y¹/y⁴ = y^-3 (Quotient of Powers)= 1/y³ (Negative Exponent)

• Put the pieces back together.
  7/-5 * x²* 1/y³
  7x²/-5y³

2) (5x⁵)³ = 125x¹⁵

• Can anything be done in the parentheses? No. Start with the Power of a Product rule.
  5³ * x^5*3 (Power of a Product)
  5³ * x¹⁵ (Power of a Power)

• Simplify.
  125x¹⁵

3) xy⁴z¹⁰ ÷ x³yz⁵ = y³ z⁵/x²

• Break down the exercise into pieces. Match constants with constants and variables with variables.
  x¹/x³ * y⁴/y¹ * z¹⁰/z⁵

• Whenever a variable doesn't have an exponent, automatically place a 1 in place of the exponent.
  x¹/x³ * y⁴/y¹ * z¹⁰/z⁵

• Simplify each piece.
  x¹/x³ = x^-2 (Quotient of Powers) = 1/x² (Negative Exponent)
  y⁴/y¹ = y³ (Quotient of Powers)
  z¹⁰/z⁵ = z⁵ (Quotient of Powers)

• Put the pieces back together.
  1/x² * y^{3 *} z⁵
  1y³ z⁵/x²
  y³ z⁵/x²

4) -42a²b⁵c³ ÷ 3a⁰b³c² = -14a²b²c

• Break down the exercise into pieces. Match constants with constants and variables with variables.
  -42/3 * a²/a⁰ * b⁵/b³ * c³/c²

• Simplify each piece.
  -42/3= -14
  a²/a⁰= a² (Quotient of Powers)
  b⁵/b³ = b² (Quotient of Powers)
  c³/c²= c¹ (Quotient of Powers)

• Put the pieces back together.
  -14a²b²c¹
  -14a²b²c

5) -56c^-6 ÷ -7c² = 8/c⁸

• Break down the exercise into pieces. Match constants with constants and variables with variables.
  -56/-7 *c^-6/c²

• Simplify each piece.
  -56/-7 = 8
  c^-6/c² = c^-8 (Quotient of Powers) = 1/c⁸ (Negative Exponent)

• Put the pieces back together.
  8 * 1/c8
  8/c⁸

6) a⁵b⁸ ÷ a⁴b⁷ = ab

• Break down the exercise into pieces. Match constants with constants and variables with variables.
  a⁵/a⁴ * b⁸/b⁷

• Simplify each piece.
  a⁵/a⁴ = a¹ (Quotient of Powers)
  b⁸/b⁷= b¹ (Quotient of Powers)

• Put the pieces back together:
  a¹*b¹
  ab

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