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📘intermediate algebra review

key term - Multiplicative Identity Property

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Definition

The multiplicative identity property states that for any real number, multiplying it by 1 will result in the original number. This property is a fundamental concept in the study of real numbers and their operations.

5 Must Know Facts For Your Next Test

  1. The multiplicative identity property is denoted by the symbol $1$, and it states that for any real number $a$, $a \times 1 = a$.
  2. This property is crucial in simplifying algebraic expressions and solving equations, as it allows for the removal or addition of the multiplicative identity without changing the value of the expression.
  3. The multiplicative identity property is one of the four basic properties of real numbers, along with the commutative, associative, and distributive properties.
  4. The multiplicative identity property, combined with the multiplicative inverse property, allows for the division of real numbers, as $a \div b = a \times b^{-1}$.
  5. Understanding the multiplicative identity property is essential for mastering concepts such as fractional arithmetic, solving linear equations, and working with algebraic expressions.

Review Questions

  • Explain how the multiplicative identity property can be used to simplify algebraic expressions.
    • The multiplicative identity property states that for any real number $a$, $a \times 1 = a$. This means that multiplying an algebraic expression by 1 will not change the value of the expression. This property can be used to simplify expressions by removing or adding the multiplicative identity, $1$, without altering the overall value. For example, if we have the expression $5x + 3$, we can multiply it by $1$ to get $5x + 3 = 5x + 3 \times 1$, which can be further simplified to $5x + 3$. This property is particularly useful when working with fractions or complex expressions, as it allows for the removal of unnecessary multiplicative factors.
  • Describe how the multiplicative identity property is used in solving linear equations.
    • The multiplicative identity property is crucial in solving linear equations, as it allows for the isolation of variables by performing inverse operations. When solving an equation such as $2x = 6$, we can divide both sides by 2 to get $x = 3$. This is possible because of the multiplicative identity property, which states that $2 \times \frac{1}{2} = 1$. By multiplying both sides of the equation by the multiplicative inverse of 2, which is $\frac{1}{2}$, we are effectively multiplying by 1, leaving the variable term isolated on one side of the equation. This property is essential in solving more complex linear equations, where the variable may be multiplied by various coefficients that need to be canceled out.
  • Analyze how the multiplicative identity property is related to the concept of multiplicative inverses.
    • The multiplicative identity property is closely linked to the concept of multiplicative inverses. The multiplicative identity property states that for any real number $a$, $a \times 1 = a$. This means that 1 acts as the multiplicative identity, as multiplying any real number by 1 results in the original number. The multiplicative inverse of a non-zero real number $a$ is the number $b$ such that $a \times b = 1$. In other words, the multiplicative inverse of a number is the value that, when multiplied by the original number, results in the multiplicative identity of 1. This relationship between the multiplicative identity property and multiplicative inverses is crucial in understanding concepts such as fractional arithmetic and solving equations involving division.