key term - Multiplication

Definition

Multiplication is a mathematical operation that involves the repeated addition of a number to itself. It is used to find the total amount or quantity when a number is multiplied by another number. Multiplication is a fundamental concept that is essential in the context of simplifying rational exponents.

5 Must Know Facts For Your Next Test

1. Multiplication is the inverse operation of division, and the two operations are closely related in the context of rational exponents.
2. Raising a number to a rational exponent is equivalent to taking the $n$th root of that number, where $n$ is the denominator of the rational exponent.
3. The laws of exponents, such as $a^{m/n} = \sqrt[n]{a^m}$, rely on the properties of multiplication to simplify and manipulate rational exponents.
4. Multiplication is used to calculate the value of expressions involving rational exponents, such as $2^{3/2}$ or $(x^2)^{1/4}$.
5. The order of operations, including the use of parentheses, is crucial when simplifying expressions with rational exponents that involve multiplication.

Review Questions

• Explain how the properties of multiplication are used to simplify expressions with rational exponents.
  • The properties of multiplication are essential in simplifying expressions with rational exponents. For example, the law of exponents $a^{m/n} = \sqrt[n]{a^m}$ allows us to rewrite expressions involving rational exponents in terms of roots, which can then be simplified using the properties of multiplication. Additionally, the order of operations, including the use of parentheses, is crucial when simplifying expressions with rational exponents that involve multiplication.
• Describe the relationship between multiplication and the concept of taking a root, as it relates to rational exponents.
  • Raising a number to a rational exponent is equivalent to taking the $n$th root of that number, where $n$ is the denominator of the rational exponent. This relationship is based on the properties of multiplication, where raising a number to a power is the inverse operation of taking a root. For example, $a^{3/2}$ is the same as $\sqrt[2]{a^3}$, which involves multiplying $a$ by itself three times and then taking the square root of the result.
• Analyze how the laws of exponents, which rely on the properties of multiplication, can be used to simplify expressions involving rational exponents.
  • The laws of exponents, such as $a^{m/n} = \sqrt[n]{a^m}$, are based on the properties of multiplication and can be used to simplify expressions involving rational exponents. By rewriting expressions with rational exponents in terms of roots, which are equivalent to raising a number to a power, the properties of multiplication can be applied to simplify the expression. This allows for the efficient manipulation and calculation of expressions with rational exponents, which is a crucial skill in the context of 8.3 Simplify Rational Exponents.

Related terms

Exponent:

An exponent is a mathematical notation that represents the number of times a base number is multiplied by itself.

Rational Number: A rational number is a number that can be expressed as a fraction, where the numerator and denominator are both integers.

Power: A power is the result of multiplying a number by itself a certain number of times, where the number of times is represented by the exponent.