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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

Perform the indicated operations. $$\left(1.2 x^{2}+3.5 x-9.6\right)+\left(2.9 x^{2}-6.7 x-3.5\right)$$

Perform the indicated operations. $$\left(-5 a^{2}-9 a^{3}+10 a^{4}\right)+\left(2 a^{3}-a^{4}\right)-\left(8 a+4 a^{3}+10 a^{4}\right)$$

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{2}$$

Simplify the expression. $$\left(-4 a^{5}\right)^{2}(a b)^{3}$$

Simplify. Write the answers with positive exponents only. $$8^{2} 8^{-4}$$
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