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Chapter 5: Problem 114

Simplify each expression using the Product Property. $$ x^{4} \cdot x^{-2} \cdot x^{-3} $$

Short Answer

Expert verified
The simplified expression is \(x^{-1}\) or \(\frac{1}{x}\).

Step by step solution

01

- Identify the Exponents

Write down the exponents of each term in the expression. For \(x^{4} \cdot x^{-2} \cdot x^{-3}\), the exponents are 4, -2, and -3.
02

- Apply the Product Property

According to the Product Property of Exponents, when multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\). So, add the exponents together: \(4 + (-2) + (-3)\).
03

- Calculate the Sum

Perform the addition of the exponents: \(4 + (-2) + (-3) = 4 - 2 - 3 = -1\).
04

- Rewrite the Expression

Using the sum found, rewrite the simplified expression as \(x^{-1}\).
05

- Final Simplification

Recognize that \(x^{-1}\) can also be written as \(\frac{1}{x}\) if further simplification is necessary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying expressions
Simplifying expressions in algebra involves reducing a potentially complex mathematical expression to its simplest form. This makes it easier to understand and work with. For example, we simplify the expression involving the variable base with different exponents as follows: start by identifying the exponents and then apply algebraic properties such as the Product Property of Exponents.
adding exponents
When simplifying expressions like the one given in the exercise, adding exponents becomes crucial. The key rule here is the Product Property of Exponents. This states that when multiplying two (or more) numbers with the same base, you can add their exponents: \(a^m \times a^n = a^{m+n}\).
So if you have \(x^4 \times x^{-2} \times x^{-3}\), you add up the exponents: \(4 + (-2) + (-3)\). This gives us \(x^{{4-2-3}} = x^{-1}\).
negative exponents
Negative exponents can often be tricky, but with the right understanding, they become simple to manage. A negative exponent denotes the reciprocal of the base raised to the positive of that exponent:\(a^{-n} = \frac{1}{a^n}\).
In our expression \(x^{-1}\), according to the rule, it can be rewritten as \(\frac{1}{x}\). This is why after simplifying \(x^4 \times x^{-2} \times x^{-3}\) to \(x^{-1}\), you can further express it as \(\frac{1}{x}\).
algebraic simplification
Algebraic simplification involves several steps to transform an algebraic expression into its simpler form. First, identify any common bases and their exponents. Next, use properties of exponents (like the Product Property) to combine these terms.
In the exercise, we first identified our exponents: 4, -2, and -3. We applied the Product Property:\(a^m \times a^n = a^{m+n}\), simplifying to\(x^{-1}\). Finally, recognize possible further simplifications using properties like negative exponents to rewrite expressions more understandable formats, such as \(\frac{1}{x}\).

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