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Product to a Power Property for Exponents
The Product to a Power Property for Exponents provides a shortcut for raising a product of factors to a power. If and are real numbers and is an integer, then:
In words: to raise a product to a power, raise each factor to that power separately. The property was originally developed for whole-number exponents using the definition of repeated multiplication. Consider : by definition, this means . Grouping like factors gives . Each factor in the original product is raised to the exponent individually. The property extends to all integer exponents.
A numerical check confirms the rule: . The left side gives , and the right side gives ✓.
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Elementary Algebra @ OpenStax
Ch.6 Polynomials - Elementary Algebra @ OpenStax
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An inventory specialist uses the expression 4^5 to determine the total number of items in a multi-level storage rack. Match each component of this mathematical expression to its correct description.
A quality control technician uses the expression 10^4 to represent the number of parts tested in a specific production batch. In this algebraic notation, what is the mathematical term for the number 4?
A manufacturing technician writes out the calculation 5 * 5 * 5 * 5 to verify the total count of a parts order. This way of writing a product, where every factor is shown individually, is called ____ notation.
A shipping coordinator is recording the storage volume of a warehouse section. Instead of writing out the repeated multiplication , the coordinator uses the compact form . True or False: In mathematics, this compact way of writing a product is specifically called exponential notation.
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A technical documentation specialist is standardizing a company’s internal database. The specialist needs to convert a repeated multiplication string, such as , into a compact format for a summary report. Arrange the following steps in the correct order to define the components and successfully write the value in exponential notation.
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A technical trainer is developing a 'Math for Technicians' guide for a manufacturing company. While explaining the expression , the trainer defines the base (). Which of the following best describes the role of the base in this notation?
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Learn After
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In a warehouse management system, the capacity of a storage bin is modeled by the expression (xy)^3. According to the Product to a Power Property, which of the following is the correct way to expand this expression?
In a technical scaling formula, the expression (5x)^2 is equivalent to 5x^2 according to the Product to a Power Property.
A logistics coordinator is using the expression (xy)^4 to calculate the capacity of a shipping container. According to the Product to a Power Property, this expression is equivalent to x^4 multiplied by ____.
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A logistics analyst is auditing the mathematical models used to calculate the capacity of different storage containers in a distribution center. Match each original capacity formula with its correctly expanded form according to the Product to a Power Property for exponents.
A design technician is scaling a blueprint where a specific dimension expansion is represented by the expression . Arrange the following steps in the correct logical sequence to expand this expression based on the fundamental definition of exponents.
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A technical trainer is creating a quick-reference card for a team of data analysts. When summarizing the Product to a Power Property for exponents, which of the following definitions should be used to describe how to correctly expand an expression like ?
A structural engineer is reviewing a design document where a specific load calculation is represented by the expression , where , , and represent dimensions. According to the Product to a Power Property, which of the following is the correct way to expand this expression?
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