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Chapter 7: Problem 81

Perform the operations. Simplify, if possible. $$ b-\frac{3}{a^{2}} $$

Short Answer

Expert verified
The expression \( b - \frac{3}{a^2} \) is already in its simplest form.

Step by step solution

01

Identify the Expression

The expression given is \( b - \frac{3}{a^2} \). This expression does not need any operation within itself other than verifying if it can be simplified.
02

Analyze Each Term

The expression has two terms: \( b \) and \( -\frac{3}{a^2} \). These terms involve different variables and cannot be combined through addition or subtraction because they are not like terms.
03

Simplify, if Possible

Check if any term can be simplified further. Here, \( b \) is a standalone variable term, and \( \frac{3}{a^2} \) cannot be simplified unless specific values for \( a \) allow it. Without given values, the expression remains \( b - \frac{3}{a^2} \).
04

Conclusion

Since the expression contains no like terms and cannot be further simplified, it is already in its simplest form.

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Key Concepts

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

Variables and Terms
In algebra, variables and terms are fundamental components. A variable is a symbol, often a letter like \( b \) or \( a \), representing a number whose value is not yet determined or can change. In this expression, \( b \) and \( a^2 \) are variables. Terms are parts of an expression separated by addition or subtraction signs.

For example, in the expression \( b - \frac{3}{a^2} \), \( b \) and \(-\frac{3}{a^2} \) are terms.
  • \( b \) is a simple term composed of the variable \( b \).
  • \(-\frac{3}{a^2}\) is a more complex term that includes the constant \(-3\) and variable expression \(a^2\) in the denominator.
Understanding variables and terms helps in identifying how parts of an expression interact and whether operations can be performed between them.
Simplifying Expressions
Simplifying expressions involves reducing them to their most straightforward form. This process can include combining like terms, reducing fractions, or clearing parentheses. However, not all expressions can be simplified significantly without additional information or context.

In the expression \( b - \frac{3}{a^2} \), simplification is limited because:
  • There are no like terms to combine. \( b \) and \( -\frac{3}{a^2} \) are distinct, with no similar variables or exponents.
  • \( \frac{3}{a^2} \) cannot be simplified further unless there is a specific value provided for \( a \).
Hence, expressions like this one might be already in their simplest form unless further numerical values are introduced. Learning to recognize this can save time and avoid unnecessary steps.
Like Terms
Like terms are terms in an algebraic expression that have the same variables raised to the same power, although the coefficients can be different.

For example, \( 2x \) and \( 5x \) are like terms because they both contain the variable \( x \) to the first power. Identifying like terms is key to simplifying expressions since you can combine them by adding or subtracting their coefficients.
  • In the expression \( b - \frac{3}{a^2} \), there are no like terms as \( b \) is distinct from \(-\frac{3}{a^2} \), given their different variables and structures.
  • Since the terms have different variables and powers, they cannot be combined.
Knowing how to identify and work with like terms is crucial for simplifying expressions efficiently and correctly.

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For each of the following purchases, determine the better buy. Memory for a computer: 128 megabytes for \(\$ 26\) or 512 megabytes for \(\$ 110\)

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