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Chapter 1: Problem 42

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

Short Answer

Expert verified
\(\frac{64}{9}\)

Step by step solution

01

- Substitute known values

First, substitute the given values of \(a\) and \(b\) into the expression. The expression is \(\left(\frac{a}{b} + a\right)^{a}\). With \(a = 2\) and \(b = 3\), it becomes \(\left(\frac{2}{3} + 2\right)^{2}\).
02

- Simplify the inner expression

Next, simplify the inner part of the expression \(\frac{2}{3} + 2\). Convert 2 to a fraction with the same denominator:\(\frac{2}{3} + \frac{6}{3} = \frac{8}{3}\).
03

- Raise the simplified expression to the power of \(a\)

Now, substitute back the simplified inner expression \(\frac{8}{3}\) into the outer expression, raising it to the power of \(2\):\[\left(\frac{8}{3}\right)^2\].
04

- Calculate the final value

Finally, compute \(\left(\frac{8}{3}\right)^2\) by squaring the numerator and the denominator individually:\[\left(\frac{8}{3}\right)^2 = \frac{8^2}{3^2} = \frac{64}{9}\].

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Key Concepts

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

Substitution in Algebra
Substitution in algebra involves replacing variables with their given values. It's an essential skill because it allows us to evaluate expressions accurately. In this exercise, we had to evaluate \(\frac{a}{b} + a\). To start, we replaced the given variables with their values. Given \(a = 2\) and \(b = 3\), the expression became: \(\frac{2}{3} + 2\). This step is important to ensure we are working with actual numbers instead of abstract variables. Always double-check the values to prevent mistakes.
Simplifying Fractions
Simplifying fractions is the process of making a fraction as simple as possible. This typically means converting it to its lowest terms, but it can also mean performing the arithmetic to combine fractions. In this exercise, we simplified \(\frac{2}{3} + 2\). To do this, we converted 2 to a fraction with a common denominator: \( 2 = \frac{6}{3} \). Then, we added \(\frac{2}{3}\) and \(\frac{6}{3}\) to get: \(\frac{8}{3}\). This step is crucial for simplifying the algebraic expressions further. Simplifying fractions often involves breaking down the problem into manageable parts.
Exponents in Algebra
Exponents in algebra signify repeated multiplication of a base number. They are a compact way to express large numbers or repetitive calculations. In this exercise, we raised \(\frac{8}{3}\) to the power of 2, meaning we multiplied \(\frac{8}{3}\) by itself. Mathematically, this is written as: \(\frac{8}{3} \times \frac{8}{3} = \frac{64}{9}\). Exponents follow specific rules, such as the power of a product rule and the power of a quotient rule. Understanding these rules helps simplify expressions quickly and accurately. To recap: \(\frac{8}{3}\)^2 = \(\frac{8^2}{3^2}\) = \(\frac{64}{9}\).

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