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Chapter 1: Problem 69

Write an equivalent expression with negative exponents. $$ \frac{1}{3 y^{4}} $$

Short Answer

Expert verified
3^{-1} y^{-4}

Step by step solution

01

Identify the components

The given expression is \(\frac{1}{3 y^{4}}\). It includes a constant in the numerator (1) and both a constant (3) and a power of \(y\) (\(y^4\)) in the denominator.
02

Apply the negative exponent rule

To rewrite the denominator with negative exponents, recall that \(\frac{1}{a^n} = a^{-n}\). Here, identify \(a = 3y^4\). Using the negative exponent rule for each term gives \(3^{-1} y^{-4}\).
03

Combine the components

Now put together the terms with negative exponents, resulting in \(3^{-1} y^{-4}\).

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Key Concepts

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

equivalent expressions
An equivalent expression in algebra means that two expressions are equal and yield the same result, even if they look different. This concept is crucial when working with negative exponents. For example, when given \(\frac{1}{3y^4}\), its equivalent expression with negative exponents is \(3^{-1} y^{-4}\). Even though the two forms look different, they are mathematically the same. Understanding how to find equivalent expressions helps simplify problems and makes them easier to work with.
exponent rules
Exponent rules are essential for simplifying expressions in algebra. One key rule is the negative exponent rule. According to this rule, any non-zero number raised to a negative exponent is equal to its reciprocal raised to the corresponding positive exponent. Mathematically, \[a^{-n} = \frac{1}{a^n} \] and vice versa. In our example, \(\frac{1}{3y^4}\) is transformed to \(3^{-1} y^{-4}\). Other important exponent rules include:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m \times n}\)
  • Power of a Product: \((ab)^n = a^n b^n\)
  • Zero Exponent: \(a^0 = 1 (a eq 0)\)
These rules help in the process of algebraic manipulation and solving expressions more efficiently.
algebraic manipulation
Algebraic manipulation involves using mathematical operations and properties to simplify expressions or solve equations. It's like rearranging pieces of a puzzle to see the bigger picture more clearly. In the context of our problem:
  • Step 1: Identify the components
    The given expression is \(\frac{1}{3 y^{4}}\)
  • Step 2: Apply the negative exponent rule
    Recall that \(\frac{1}{a^n} = a^{-n}\). Here, we apply it to each term in the denominator.
  • Step 3: Combine the components
    This gives us \(3^{-1} y^{-4}\), an equivalent expression with negative exponents.
Mastering algebraic manipulation helps in solving complex algebraic equations, simplifying expressions, and understanding the structure of mathematical formulas. Practicing these steps regularly makes these processes more intuitive and manageable.

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