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Chapter 5: Problem 20

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

Short Answer

Expert verified
27

Step by step solution

01

Substitute Given Values

Substitute the value of 2 for a and 3 for b in the expression \(9 \cdot \frac{1}{b}\)^{b}.\ The expression becomes \(9 \cdot \frac{1}{3}\)^{3}.
02

Simplify the Expression Inside the Parentheses

Simplify the expression inside the parentheses: \(9 \cdot \frac{1}{3} = 3\). So the expression now looks like this: \(3^{3}\).
03

Title: Calculate the Power

Raise 3 to the power of 3: \(3^{3} = 27\).

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Key Concepts

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

Substituting Variables
To start solving problems involving algebraic expressions, it is crucial to understand substituting variables. This means you replace the variables in an expression with their given values.

In our exercise, we substitute the value of 2 for **a** and 3 for **b**. The expression given is \((9 \backslash cdot \frac{1}{b})^{b}\). When we substitute **b** with 3, the expression becomes \((9 \backslash cdot \frac{1}{3})^{3}\).

By substituting variables correctly, we can transform more abstract algebraic expressions into something more concrete and easier to handle.
Exponents
Exponents represent repeated multiplication. The expression \(x^{n}\) means multiplying **x** by itself **n** times. For example, \(3^{3}\) means \(3 \backslash cdot 3 \backslash cdot 3\), which equals 27.

After substituting the variables in our expression, we got \((3)^{3}\). Here, the exponent tells us to multiply three, three times: \3 \backslash cdot 3 \backslash cdot 3 = 27\.

Understanding how exponents work is essential for correctly solving expressions involving powers.
Simplifying Expressions
Simplifying expressions involves combining like terms and using arithmetic to reduce the expression to its simplest form.

In our example, we start with the expression \(9 \backslash cdot \frac{1}{3} \backslash ^{3}\). The first step is to simplify the part inside the parentheses: \(9 \backslash cdot \frac{1}{3} = 3\). This step transforms the initial complex expression into a more straightforward one, \(3^{3}\).

Always simplify inside the parentheses first, following the order of operations, to make the expression easier to evaluate.
Evaluating Expressions
Evaluating expressions involves performing arithmetic operations to find the expression's value. This is the final step after substituting variables and simplifying the expression.

In our example, after simplification, we have \(3^{3}\). Evaluating this means calculating the value of 3 raised to the power of 3, which is 27.

By carefully going through each step 鈥� substituting, simplifying, and then evaluating 鈥� you can accurately solve algebraic expressions.

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Most popular questions from this chapter

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