Laws of Exponents

Definition

The laws of exponents are a set of rules that govern the manipulation and simplification of expressions involving exponents. These laws provide a systematic way to perform operations with powers and exponential expressions, which are essential in the study of intermediate algebra, particularly in the context of properties of exponents and scientific notation.

5 Must Know Facts For Your Next Test

1. The product of powers with the same base is the power with the exponents added.
2. The quotient of powers with the same base is the power with the exponents subtracted.
3. The power of a power is the power with the exponents multiplied.
4. The power of a product is the product of the powers.
5. The power of a quotient is the quotient of the powers.

Review Questions

• Explain how the laws of exponents can be used to simplify expressions involving multiplication and division of powers with the same base.
  • The laws of exponents state that when multiplying powers with the same base, the exponents are added, and when dividing powers with the same base, the exponents are subtracted. For example, $a^3 \cdot a^4 = a^{3+4} = a^7$, and $a^8 \div a^5 = a^{8-5} = a^3$. These rules allow for the efficient simplification of expressions involving the multiplication and division of powers with the same base.
• Describe how the laws of exponents can be applied to expressions involving the power of a power.
  • The laws of exponents state that the power of a power is the power with the exponents multiplied. For example, $(a^3)^4 = a^{3 \cdot 4} = a^{12}$. This rule allows for the simplification of expressions where a power is raised to another power, by multiplying the exponents. This is a useful property in working with scientific notation and other applications involving exponents.
• Analyze how the laws of exponents can be used to simplify expressions involving the power of a product or quotient.
  • The laws of exponents state that the power of a product is the product of the powers, and the power of a quotient is the quotient of the powers. For example, $(a \cdot b)^3 = a^3 \cdot b^3$, and $(a \div b)^4 = a^4 \div b^4$. These rules allow for the efficient simplification of expressions where a product or quotient is raised to a power, by distributing the exponent to the individual factors. This is an important skill in manipulating complex exponential expressions, particularly in the context of scientific notation.

Related terms

Exponent: A number that indicates how many times a base number is multiplied by itself.

Power: An expression that represents repeated multiplication of a number by itself.

Base: The number that is being raised to a power or exponent.