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Chapter 10: Problem 31

Simplify. Write the answer with positive exponents only. $$a^{-4}$$

Short Answer

Expert verified
\(\frac{1}{a^4}\)

Step by step solution

01

Identify the Negative Exponent

Identify that the given expression has a negative exponent. The expression is \(a^{-4}\).
02

Apply the Property of Negative Exponents

Use the property that \(a^{-n} = \frac{1}{a^n}\) for any non-zero number \(a\). Therefore, \(a^{-4}\) can be rewritten as \(\frac{1}{a^4}\).
03

Write with Positive Exponent

Rewrite the expression with a positive exponent. The final simplified expression is \(\frac{1}{a^4}\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a key skill in algebra. It involves reducing an expression to its simplest form. This means using algebraic properties to combine like terms and simplify operations.

Let's break down the process:
  • Identify like terms
  • Combine these terms
  • Use algebraic properties to simplify
In our example, the expression is \(a^{-4}\). This expression can be simplified by addressing the negative exponent. By doing this, we convert the given expression into an easier-to-understand form.
Positive Exponents
Positive exponents indicate the number of times a base is multiplied by itself. For example, \(a^3\) means \(a\) is multiplied by itself three times: \(a \times a \times a\).
Negative exponents work differently. When you see a negative exponent, you can use the rule \(a^{-n} = \frac{1}{a^n}\), which means the base is on the opposite side of the fraction bar.

In our given problem, \(a^{-4}\) becomes \(\frac{1}{a^4}\) when the negative exponent rule is applied. So instead of multiplying \(a\) four times, we divide 1 by \(a\) four times.
Algebraic Properties
Algebraic properties are rules that help simplify and solve equations. Some key properties include:
  • Commutative Property: Order doesn't matter (\(a + b = b + a\))
  • Associative Property: Grouping doesn't matter (\((a + b) + c = a + (b + c)\))
  • Distributive Property: Distributing multiplication over addition (\(a(b + c) = ab + ac\))
  • Exponent Properties: Such as \(a^m \cdot a^n = a^{m+n}\) and \(a^2 / a^3 = a^{2-3}\)
In solving the given exercise, we specifically used the property of negative exponents to turn \(a^{-4}\) into a fraction \(\frac{1}{a^4}\). Knowing and applying these properties make algebra simpler and more predictable.

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Most popular questions from this chapter

For Exercises \(3-12\), simplify the expression. Write the answers with positive exponents only. $$\frac{z^{10} z^{20}}{z^{5}}$$

Write the opposite of the polynomial. $$2.4 m-5.1 n$$

Multiply. $$\left(-8 m n^{5}\right)\left(5 m^{2} n^{3}\right)$$

$$\text { Fill in the blanks }\left(x^{3}\right)^{5}=x \square \cdot \square=x^{15}$$

For Exercises \(65-72\), write the mass of the planet in scientific notation if it is not already in scientific notation. (TABLE CAN'T COPY) $$\text { Saturn } \quad 586.5 \times 10^{24}$$
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