Pre-Algebra - Multiplicative Inverse Property

Which of the following displays the multiplicative inverse property?

$a \cdot \frac{1}{a} = 1$

$a \cdot 1 = a$

$a \cdot a = a^2$

$a \cdot -1 = -a$

$a \cdot 0 = 0$

Explanation

The mulitplicative inverse property deals with reciprocals. For example, the multiplicative inverse, or reciprocal, of the number 7 is $\frac{1}{7}$ .

The multiplicative inverse property states that a number times its multiplicative inverse equals 1.

Therefore,

$a \cdot \frac{1}{a} = 1$

displays the multiplicative inverse property.

Which of the following statements demonstrates the inverse property of multiplication?

$\frac{6}{7} \times \frac{7}{6} = 1$

$\frac{6}{7} \times 1 = \frac{6}{7}$

$\frac{6}{7} \times\left ( \frac{1}{4} \times \frac{3}{8} \right ) =\left ( \frac{6}{7} \times\frac{1}{4} \right ) \times \frac{3}{8}$

$\frac{6}{7} \times\frac{1}{4} = \frac{1}{4} \times \frac{6}{7}$

None of the examples in the other responses demonstrates the inverse property of multiplication.

Explanation

The inverse property of multiplication states that for every real number, a number exists, called the multiplicative inverse, such that the number and its inverse have product 1. Of the statements given, only

$\frac{6}{7} \times \frac{7}{6} = 1$

demonstrates this property.

Simplify.

$\frac{\frac{12}{c}}{\frac{3}{c^2}}$

$\frac{4}{c}$

$\frac{1}{4c}$

$\frac{3}{4c^2}$

Explanation

To simplify a compound fraction, multiply the numerator by the reciprocal of the denominator. Remember that a compound fraction can easily be rewritten as a division problem!

$\frac{\frac{12}{c}}{\frac{3}{c^2}}=\frac{12}{c}\div\frac{3}{c^2}=\frac{12}{c}\times\frac{c^2}{3}$