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Chapter 5: Problem 21

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

Short Answer

Expert verified
The value of the expression is 1.

Step by step solution

01

Substitute the given values

First, plug in the values for 饾憥 and 饾憦 into the expression. Given that 饾憥 = 2 and 饾憦 = 3, the expression \(\frac{b}{a} \times \frac{a}{b} \) becomes \(\frac{3}{2} \times \frac{2}{3} \).
02

Simplify the fraction multiplication

Next, simplify the product of the fractions \(\frac{3}{2}\times \frac{2}{3}\). When multiplying fractions, multiply the numerators together and the denominators together. \[ \frac{3 \times 2}{2 \times 3} = \frac{6}{6} = 1 \]
03

Evaluate the exponentiation

Now, evaluate the expression with the simplified product. The expression is now \(1^a\), where 饾憥 = 2. Therefore, \(\boxed{1^2}\) is equal to \(\boxed{1}\).

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Key Concepts

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

substitute values
Substituting values is a fundamental skill in algebra. It involves replacing variables in an expression with given numbers.
To achieve this:
  • Identify the variables in your expression.
  • Replace each variable with the provided value.
For example, in the expression \(\frac{b}{a} \times \frac{a}{b} \), where 饾憥 = 2 and 饾憦 = 3:
Replace 饾憥 with 2 and 饾憦 with 3:
\(\frac{3}{2} \times \frac{2}{3} \).
This step lays the groundwork for simplifying the expression.
simplify fractions
Simplifying fractions involves combining and reducing them to their simplest form.
When multiplying fractions:
  • Multiply the numerators together.
  • Multiply the denominators together.
For example, simplify \(\frac{3}{2} \times \frac{2}{3} \):
Multiply the numerators: 3 脳 2 = 6
Multiply the denominators: 2 脳 3 = 6
Thus, \(\frac{3 \times 2}{2 \times 3} = \frac{6}{6} = 1\).
Always simplify the result if possible. In this case, 1 is already in its simplest form.
exponentiation
Exponentiation involves raising a number (the base) to a power (the exponent).
The notation for exponentiation is \(a^n \), where:
  • 饾憥 is the base.
  • n is the exponent.
In our example, after simplifying the fraction we have 1 as the base.
Thus, we need to evaluate \(1^a\), where 饾憥 = 2:
Any number to the power of zero is 1, but in this case, we have 1 to the power of 2:
So, \(1^2 = 1 \times 1 = 1\).
Understanding exponentiation helps you simplify expressions requiring powers efficiently.

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Most popular questions from this chapter

Consider what happens when you rotate a linear graph \(180^{\circ} .\) a. Graph the line \(y=2 x+4\) b. On the same grid, draw the image of the line under a \(180^{\circ}\) rotation centered at the origin. c. What do you notice about the image line and the original line? d. Write an equation of the new line. e. Does your equation in Part d support your observation in Part c? Explain.

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