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Chapter 10: Problem 364

In the following exercises, simplify. $$\left(a^{3}\right)^{-3}$$

Short Answer

Expert verified
\left( a^3 \right)^{-3} = \frac{1}{a^9} \

Step by step solution

01

Identify the Expression

The given expression is \( \left( a^3 \right)^{-3} \). This involves an exponent raised to another exponent and a negative exponent.
02

Apply the Power of a Power Property

Use the property of exponents \( \left( a^m \right)^n = a^{m \cdot n} \. \) Apply this to the expression: \( \left( a^3 \right)^{-3} = a^{3 \cdot -3} \). Simplifying the exponent, we get: \( a^{-9} \).
03

Simplify the Negative Exponent

To simplify \( a^{-9} \,\) remember that a negative exponent means the reciprocal of the base with a positive exponent. Therefore, \ a^{-9} = \frac{1}{a^9} \.

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Key Concepts

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

Negative Exponents
In mathematics, negative exponents can seem tricky, but they are actually quite simple when you break them down. A negative exponent indicates that the base should be taken as the reciprocal. For example, if you see the expression \(a^{-n}\), it means the same as \(\frac{1}{a^n}\). This property helps us simplify expressions with negative powers.

When solving problems involving negative exponents, just remember these key points:
  • A negative exponent moves the base to the denominator of a fraction (or numerator if it鈥檚 already in the denominator).
  • Once the base is in the reciprocal form, change the exponent to a positive.
By applying these steps, you can simplify expressions correctly and efficiently.
Power of a Power Property
The power of a power property is a rule used to simplify expressions where an exponent is raised to another exponent. This property is stated as: \((a^m)^n = a^{m \cdot n}\). This means you multiply the exponents together.

Using this rule makes the simplification process straightforward. For instance, consider the expression \( (a^3)^{-3} \). Here, \(a^3\) is raised to the power of \(-3\). According to the power of a power property, we can rewrite it as \( a^{3 \cdot -3} \), which simplifies to \(a^{-9}\).

Just follow these steps to apply this rule properly:
  • Identify the base and the exponents.
  • Multiply the exponents together.
  • Simplify the result.
By using this property, complex expressions can be made much simpler to handle.
Reciprocal
A reciprocal is a way of converting expressions with negative exponents into more manageable forms. The reciprocal of a number is 1 divided by that number. For instance, the reciprocal of \(a\) is \(\frac{1}{a}\). This concept is essential in simplifying negative exponents.

When you see an expression like \(a^{-9}\), using the reciprocal concept, you can rewrite it as \(\frac{1}{a^9}\). This conversion helps in understanding and solving the expression more easily.

Here鈥檚 a quick guide to working with reciprocals:
  • Identify the term with the negative exponent.
  • Rewrite the term as the reciprocal with a positive exponent.
  • Simplify further if needed.
By mastering the reciprocal concept, handling negative exponents becomes intuitive and much less daunting.

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Most popular questions from this chapter

Simplify. $$\left(k^{-2}\right)^{-3}$$

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