Problem 58 - Elementary Algebra

Question

\(\left(\frac{4 w^{3}}{q^{5}}\right)^{2}\left(\frac{w}{q}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{16 w^{7}}{q^{11}} \).

Step 1 - Distribute the Exponent

Apply the exponent 2 to both the numerator and the denominator of the fraction inside the parentheses: \( \left(\frac{4 w^{3}}{q^{5}}\right)^{2} = \frac{(4 w^{3})^2}{(q^{5})^2} = \frac{16 w^{6}}{q^{10}} \)

Step 2 - Multiply the Fractions

Multiply the resultant fraction by the second fraction: \( \left( \frac{16 w^{6}}{q^{10}} \right) \left( \frac{w}{q} \right) = \frac{16 w^{6} \cdot w}{q^{10} \cdot q} \)

Step 3 - Simplify the Expression

Combine the variables with the same base in the numerator and the denominator: \( \frac{16 w^{6+1}}{q^{10+1}} = \frac{16 w^{7}}{q^{11}} \)

Key Concepts

distributing exponents

Distributing exponents is a fundamental concept in algebra. It involves applying the exponent to both the numerator and the denominator of a fraction inside parentheses. For example, in the expression \(\text{left}(\frac{4 w^{3}}{q^{5}}\text{right}){^2}\), we distribute the exponent 2 to each part of the fraction: the constants, and the variables. This means we calculate \((4 w^3)^2\) and \((q^5)^2\) separately.

To do this:

• First, square the constant 4, resulting in 16.
• Next, apply the exponent to \(w^3\), giving \(w^{6}\).\t
• Lastly, square \(q^5\), giving \(q^{10}\).

Combining these results, we get \(\frac{16 w^{6}}{q^{10}}\). This technique is key to handling more complex algebraic expressions involving fractions and exponents.

multiplying fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. For instance, in the exercise, after distributing the exponents, we need to multiply \(\text{left}\frac{16 w^{6}}{q^{10}}\text{right}\) by \( \text{left}\frac{w}{q}\text{right} \).

To do this:

• Multiply the numerators \(16 w^{6} \times w\), resulting in \((16 w^{7})\).
• Multiply the denominators \(q^{10} \times q\), resulting in \((q^{11})\).

This process ensures that we correctly combine the fractions.

The intermediate result then is \(\frac{16 w^{7}}{q^{11}}\). This step is crucial as mistakes in multiplying the numerators and denominators can lead to incorrect results.

simplifying expressions

Simplifying expressions means combining like terms and reducing the expression to its simplest form. In our exercise, we start with \(\frac{16 w^{7}}{q^{11}}\). We need to ensure that each term in the numerator and the denominator is simplified as much as possible.

Here are the steps:

• Check for any common factors in the numerator and denominator and cancel them out.
• Ensure that exponents are correctly added when multiplying like bases (as seen in our fraction multiplication).

In this case, there are no more common factors to cancel.

So, the simplest form of our expression is \(\frac{16 w^{7}}{q^{11}}\). Simplifying expressions is a vital skill in algebra, as it makes equations easier to manage and solutions clearer.