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Chapter 5: Problem 146

Simplify each expression. $$ \left(\frac{7}{9} p q^{4}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \( \frac{49}{81} p^2 q^8 \).

Step by step solution

01

Identify the Expression Components

Recognize that the expression \(\frac{7}{9} pq^{4}\) is a product of a fraction and variables raised to their respective powers.
02

Apply the Power Rule to the Fraction

Using the power rule \(\big(\frac{a}{b}\big)^n = \frac{a^n}{b^n}\), apply the power of 2 to both the numerator and the denominator: \[ \bigg(\frac{7}{9}\bigg)^2 = \frac{7^2}{9^2} = \frac{49}{81} \]
03

Apply the Power Rule to the Variables

Apply the power of 2 to the variables by using the power rule for exponents \( (x^m)^n = x^{mn} \): \[ (pq^4)^2 = p^2 (q^4)^2 = p^2 q^8 \]
04

Combine the Results

Multiply the fraction obtained in Step 2 by the variables obtained in Step 3: \[ \frac{49}{81} p^2 q^8 \]

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Key Concepts

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

The Power Rule
In algebra, the power rule helps us manage exponents. This rule says that when you raise a power to another power, you multiply the exponents together. For example, if you have \((x^m)^n = x^{mn}\). This is helpful when simplifying expressions that involve exponents raised to another exponent.
When we applied the power rule in the exercise, we managed separately the fraction part \((\frac{7}{9})^2\) and the variables \((pq^4)^2\). Breaking it down like this makes it easier to simplify complex expressions step by step.
Understanding Exponents
Exponents are a shorthand way to show repeated multiplication. For instance, \(a^4 = a \times a \times a \times a\). When you see exponents in an expression, it means the base is multiplied by itself multiple times.
In the given problem, we have two exponents to handle: the exponent on the variable \(q\), which is initially 4, and then we raised the entire expression to the power of 2. This means we needed to apply the power rule to distribute the exponents properly.
Variables in Algebra
Variables represent unknown values and can be any letter (like \(p\) or \(q\)). They work just like numbers, following the same arithmetic rules, including those for exponents.
In our example, \(p\) was raised to the power of 2, and \(q^4\) was raised to the power of 2 as well. Applying the power rule, we got \(p^2 \) and \(q^{8}\). This shows how variables and their exponents can be manipulated using algebraic rules.
Fraction Multiplication
When it comes to fractions, multiplication is straightforward if you understand the components. Each fraction has a numerator (top part) and a denominator (bottom part). To multiply fractions, you multiply the numerators together, and the denominators together.
In the given problem, \(\frac{7}{9}\) squared becomes \(\frac{7^2}{9^2} = \frac{49}{81}\). By applying the power rule, we simplified the multiplication of the fraction under exponent 2. This step is crucial in getting the correct simplified form of the expression.

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