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Chapter 5: Problem 19

Evaluate each expression for a 2 and b 3. $$ \left(\frac{1}{9}\right)^{a} $$

Short Answer

Expert verified
\(\frac{1}{9}\)^{2} = \( \frac{1}{81} \).

Step by step solution

01

Identify the given values

The problem states to evaluate the expression \(\frac{1}{9}\)^{a} for \( a = 2 \). This means we will substitute \( a = 2 \) into the given expression.
02

Substitute the given value into the expression

Replace \( a \) with 2 in the expression \(\frac{1}{9}\)^{a} . This gives us \(\frac{1}{9}\)^{2} .
03

Evaluate the exponentiation

Evaluate \(\frac{1}{9}\)^{2} by calculating \(\frac{1}{9} \times \frac{1}{9} \). This results in \(\frac{1}{81}\).

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Key Concepts

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

Substitution in Algebra
In algebra, substitution is the process of replacing a variable with its numerical value to simplify an expression. Let's break it down with an example.
  • First, identify the values you need to substitute. In our exercise, we need to substitute the value of \( a \), which is 2, into the given expression \( \left(\frac{1}{9}\right)^{a} \).
  • Second, replace the variable \( a \) with its given value. Our expression now becomes \( \left(\frac{1}{9}\right)^{2} \).
Substitution helps us transform algebraic expressions into numerical ones, making them easier to solve. This step is crucial in progressing toward the solution.
Exponentiation
Exponentiation is a mathematical operation that involves raising a base number to a power. In our exercise, we raised the fraction \( \frac{1}{9} \) to the power 2. Here is how it works:
  • The base is \( \frac{1}{9} \).
  • The exponent is 2, which means we multiply the base by itself twice.
To visualize: \( \left(\frac{1}{9}\right)^{2} \) translates to \( \frac{1}{9} \times \frac{1}{9} \).
Simplifying this, we get \( \frac{1}{81} \).
Remember: Exponentiation with fractions follows the same rules as with whole numbers.
Fractions
Fractions represent a part of a whole and are used frequently in algebra. Understanding fractions is crucial for evaluating certain expressions. In our case, we started with the fraction \( \frac{1}{9} \).
Here are some key points:
  • The numerator is the top number (1 in \( \frac{1}{9} \)).
  • The denominator is the bottom number (9 in \( \frac{1}{9} \)).
When we raised \( \frac{1}{9} \) to the power of 2, we multiplied the fraction by itself. This involves:
  • Multiplying the numerators together: \( 1 \times 1 = 1 \).
  • Multiplying the denominators together: \( 9 \times 9 = 81 \).
This results in the fraction \( \frac{1}{81} \).
Understanding how to handle fractions is a key skill in algebra, especially when combined with exponentiation.

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