Learn Before
Simplifying Using the Inverse Property of Multiplication
Simplify the product of three fractions by recognizing a reciprocal pair and reordering factors:
Step 1 — Spot the reciprocals: The first factor and the third factor are reciprocals of each other, because their numerators and denominators are swapped.
Step 2 — Reorder using the commutative property of multiplication: Swap the second and third factors so that the reciprocal pair sits side by side:
Step 3 — Apply the inverse property of multiplication: The reciprocal pair multiplies to :
Step 4 — Apply the multiplicative identity: Multiplying by leaves the remaining fraction unchanged:
This example mirrors the addition strategy of scanning for additive inverses (opposites), but in a multiplication setting: before multiplying left to right, look for factors that are reciprocals. Rearranging them next to each other allows them to collapse to , simplifying the computation in fewer steps.
0
1
Tags
OpenStax
Elementary Algebra @ OpenStax
Ch.1 Foundations - Elementary Algebra @ OpenStax
Algebra
Math
Prealgebra
Related
Simplifying Using the Inverse Property of Multiplication
In a business spreadsheet, if a user multiplies a cell value by a non-zero constant 'k', the Inverse Property of Multiplication states that multiplying 'k' by its multiplicative inverse (reciprocal) '1/k' will yield a product of:
In a software engineering context, the Inverse Property of Multiplication states that the product of any real number and its multiplicative inverse is always 1, including the number zero.
Algorithm Scaling Verification
In a laboratory setting, a technician uses various 'scaling factors' to adjust sensor readings. To return a reading to its baseline value of 1, the technician must multiply the current factor by its multiplicative inverse. Match each scaling factor with its correct multiplicative inverse according to the Inverse Property of Multiplication.
In a digital imaging application, a technician applies a zoom factor of 4 to a photograph. To restore the image to its original size, the software multiplies the zoom factor by its multiplicative inverse (reciprocal). According to the Inverse Property of Multiplication, the product of 4 and its reciprocal is ____.
A software developer is writing a routine to 'undo' a scaling factor applied to a graphic. To correctly apply the Inverse Property of Multiplication, arrange the following logical steps in the order they occur to satisfy the property's definition.
Manufacturing Speed Adjustment Verification
Explaining Multiplication Reversal in Data Auditing
In a corporate budget report, an analyst applies a 'correction factor' to adjust for inflation. To return the adjusted figures to their original baseline, the analyst multiplies the non-zero correction factor by its multiplicative inverse (reciprocal). Which mathematical property ensures that the product of this operation is always 1?
A quality assurance engineer is developing a 'Property Validation' module for a mathematical software library. Which of the following equations should the engineer use to correctly represent the Inverse Property of Multiplication for any non-zero value 'x'?
Learn After
A technician is scaling a chemical solution using three ratios: (7/15), (8/23), and (15/7). To simplify the calculation, the technician identifies (7/15) and (15/7) as reciprocals. According to the Inverse Property of Multiplication, what is the product of these two reciprocal fractions?
A pharmacy technician is calculating a dosage adjustment using the expression (7/15) * (8/23) * (15/7). By recognizing the reciprocal pair and simplifying the calculation, the technician finds the final result to be ____.
A logistics coordinator is calculating the weight distribution of a shipment using the expression . To simplify the calculation efficiently, match each component of the process with its correct description or value.
A quality control analyst is simplifying the combined ratio . True or False: According to the Inverse Property of Multiplication, the analyst can simplify the calculation by recognizing that the product of the reciprocal pair and is 1.
A logistics analyst is calculating a weight distribution factor for a shipment using the product of three ratios: . Arrange the following steps in the correct order to simplify this calculation using the Inverse Property of Multiplication.
Reciprocal Ratios in Supply Chain Scaling
Efficiency Ratios in Production Scaling
Applying Inverse Properties to Financial Ratio Audits
A quality control analyst is using three conversion ratios: , , and . The analyst identifies and as a reciprocal pair. According to the definition of a reciprocal, what characteristic makes these two fractions a pair?
A logistics coordinator is calculating a weight distribution factor by multiplying three ratios: . To simplify the calculation using the Inverse Property of Multiplication, which two factors should the coordinator identify as a reciprocal pair?