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Questions and Answers
What does x^(-3) represent?
What does x^(-3) represent?
What is the result of x^(-3) when simplified?
What is the result of x^(-3) when simplified?
What is the outcome of x^(-3)*x^(3)?
What is the outcome of x^(-3)*x^(3)?
In the expression x^2, what does the 2 represent?
In the expression x^2, what does the 2 represent?
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How does the product rule of negative exponents apply to x^(-5)*x^(5)?
How does the product rule of negative exponents apply to x^(-5)*x^(5)?
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What is x^(-4) equal to?
What is x^(-4) equal to?
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What is the result of dividing x^5 by x^(-2)?
What is the result of dividing x^5 by x^(-2)?
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If y^(-4) is given, what is the reciprocal of y^4?
If y^(-4) is given, what is the reciprocal of y^4?
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For the function f(x) = x^(-2), what is the inverse function f^(-1)(x)?
For the function f(x) = x^(-2), what is the inverse function f^(-1)(x)?
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Which statement is true regarding negative exponents?
Which statement is true regarding negative exponents?
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If given z^(3/2), what is its equivalent form with a negative exponent?
If given z^(3/2), what is its equivalent form with a negative exponent?
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When dividing with negative exponents, what is the relationship between the exponents of the dividend and the divisor?
When dividing with negative exponents, what is the relationship between the exponents of the dividend and the divisor?
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Study Notes
The Power of Negative Exponents: Understanding x^(-3)
Negative exponents, often seen in algebraic expressions like x^(-3), might initially appear confusing, but they're actually quite simple once you understand their underlying principles. In this article, we'll take a closer look at negative exponents, breaking down what x^(-3) means and how to handle these expressions in context.
The Basics of Exponents
Let's start with a brief refresher on the basics of exponents. An expression like x^2 indicates that we should multiply the base x by itself the number of times specified by the exponent, i.e., x * x = x^2.
Negative Exponents
Now, let's consider a negative exponent. x^(-3) does not mean that we should multiply the base x by itself a negative number of times; rather, it signifies the reciprocal of x raised to the absolute value of the exponent. In other words:
x^(-3) = 1/x^3
Properties of Negative Exponents
Some helpful properties to remember when working with negative exponents include:
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Multiplication of negative exponents: If the base is the same and the exponents are opposite in sign, the products of their absolute values are equal, and their signs are opposite. For instance, x^(-3)*x^(3) = x^(3-(-3)) = x^6 and x^(-3) = 1/x^3.
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Division of negative exponents: If the base is the same and the exponent of the divisor is the opposite sign of the exponent of the dividend, then the absolute value of the exponent of the dividend minus the absolute value of the exponent of the divisor is equal to the exponent of the result. For example, 1/x^3 = x^(-3).
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Powers of negative exponents: If the exponent is negative, the power of that exponent is the reciprocal of the base raised to the absolute value of the exponent. For instance, x^(-3) = 1/x^3 and x^(-1/2) = 1/√x.
Applications of Negative Exponents
Negative exponents are useful in various contexts, particularly when dealing with inverse functions. For example, if f(x) = x^n, then the inverse function f^(-1)(x) = x^(1/n).
Common Misconceptions
While negative exponents may initially seem unusual, there are no special rules governing their use. They follow the same rules as positive exponents, and they are simply a convenient way to express certain algebraic relationships.
Conclusion
Negative exponents, like x^(-3), may seem strange at first glance, but they're actually quite straightforward. They follow the same rules as positive exponents, just with a different interpretation. By understanding the basic principles underlying negative exponents, you'll be able to handle these expressions with ease, making your algebraic work a breeze.
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Description
Explore the concept of negative exponents, such as x^(-3), in algebra and learn how to interpret and manipulate expressions with negative exponents. This article covers the basics of exponents, the properties of negative exponents, applications in inverse functions, and common misconceptions to help you grasp this fundamental algebraic concept.
