Math - Laws of Exponents

General Note Details

• Subject: Math-Laws of Exponets
• Date: 9/10/26

Definitions and Conceptual Descriptions of Exponent Laws

• Product of Powers:

  • Definition: When multiplying two powers with the same base, add the exponets.
  • Detailed Explanation: When two exponential expressions sharing the identical base are multiplied together, the result is obtained by maintaining the common base and taking the sum of the exponents.

• Quotient of Powers:

  • Definition: When dividing two Powers with the same base, Subtract the exponets.
  • Detailed Explanation: When dividing an exponential expression by another with the exact same base, the result is calculated by keeping the shared base and subtracting the exponent of the denominator from the exponent of the numerator.

• Power of a Power:

  • Definition: When raising a Power to another Power, multiply the exponets.
  • Detailed Explanation: When an expression already raised to a power is raised to an additional outer exponent, the simplified term keeps the base and multiplies the inner and outer exponents together.

• Power of a Product:

  • Definition: When raising a product to a Power, apply the exponet to each part.
  • Explicit Formula Mentioned: $(ab)^n = a^n \cdot b^n$
  • Detailed Explanation: When a multi-factor term inside parentheses is raised to a single power, that exponent must be applied individually to every factor within the product.

• Power of a Quotient:

  • Definition: When raising a quotient to a Power, you apply the exponet to the numerator and denominator.
  • Detailed Explanation: Raising a fractional term to an exponent requires distributing that exponent to both the top component (numerator) and bottom component (denominator) of the quotient.

• Zero Exponent:

  • Definition: Any non-zero base raised to $0$ is one.
  • Detailed Explanation: For any non-zero real base $a$, raising the base to the power of zero always equals $1$.

• Negative Exponent:

  • Definition: Indicates the reciprocal of the base raised to the absolute value of the exponet.
  • Detailed Explanation: A negative value in an exponent signifies taking the multiplicative inverse (reciprocal) of the base and changing the sign of the exponent to its positive absolute value.

• Fractional Exponents:

  • Definition: This indicates a root (noted in source as "noot").
  • Detailed Explanation: Exponents expressed as fractions correspond to radical expressions, where the denominator of the fraction acts as the root index and the numerator acts as the power to which the base is raised.

Summary of Exponent Rules and Formulas

• Product of Powers:

  • Rule: Product of Powers
  • Formula: $a^m \cdot a^n = a^{m+n}$

• Quotient of Powers:

  • Rule: Quotient of Powers
  • Formula: $\frac{a^m}{a^n} = a^{m-n}$

• Power of a Power:

  • Rule: Power of a Power
  • Formula: $(a^m)^n = a^{m \cdot n}$

• Power of Product:

  • Rule: Power of Product
  • Formula: $(ab)^m = a^m \cdot b^m$

• Power of Quotient:

  • Rule: Power of Quotient
  • Formula: zás$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

• Zero Exponent:

  • Rule: Zero Exponent
  • Formula: $a^0 = 1$

• Negative Exponent:

  • Rule: Negative Exponent
  • Formula: $a^{-n} = \frac{1}{a^n}$

• Fractional Exponent:

  • Rule: Fractional Exponent
  • Formula: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$