Problem 48 - Basic College Mathematics

Question

Write the answer using fraction notation. $$ \left(\frac{57}{61}\right)^{3} $$

Step-by-Step Solution

Answer

\(\frac{185193}{227881}\)

Step 1: Identify the Base and the Exponent

Recognize that the given expression is in the form of a fraction raised to a power. Here, the base is the fraction \(\frac{57}{61}\), and the exponent is 3.

Step 2: Apply the Exponent to the Numerator and Denominator

In a fraction \(\frac{a}{b}\) raised to a power \, both the numerator \(a\) and the denominator \(b\) are raised to the power individually. So \(\bigg(\frac{57}{61}\bigg)^3 = \frac{57^3}{61^3}\).

Step 3: Compute the Numerator

Calculate \(57^3\). \(57 \times 57 = 3249\) and \(3249 \times 57 = 185193\). Therefore, \(57^3 = 185193\).

Step 4: Compute the Denominator

Calculate \(61^3\). \(61 \times 61 = 3721\) and \(3721 \times 61 = 227881\). Therefore, \(61^3 = 227881\).

Step 5: Write the Final Fraction

Combine the results from steps 3 and 4 to write the final fraction: \(\frac{57^3}{61^3} = \frac{185193}{227881}\).

Key Concepts

exponents

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. In the given problem, we are working with the exponent 3. This means the base \(\frac{57}{61}\) is multiplied by itself three times. Understanding exponents is crucial because they allow us to simplify and solve expressions quickly. For example, instead of writing \(\frac{57}{61} \times \frac{57}{61} \times \frac{57}{61}\), we write \(\bigg(\frac{57}{61}\bigg)^3\). This notation makes it easier to manipulate and understand large expressions.

numerator

The numerator is the top part of a fraction. It shows how many parts of the whole are being considered. In the given exercise, 57 is the numerator of the fraction \(\frac{57}{61}\). When we apply the exponent to the fraction, we need to raise both the numerator and the denominator to that power. Therefore, we calculate \(57^3\) to get 185193. This step ensures that the fraction maintains its proper value when raised to the power.

denominator

The denominator is the bottom part of a fraction. It indicates into how many equal parts the whole is divided. In our problem, the denominator is 61. Similar to the numerator, we apply the exponent to the denominator. We calculate \(61^3\) to get 227881. By raising both the numerator and the denominator to the power, we maintain the fraction's relationship and value.

fraction computation

Fraction computation involves the arithmetic of fractions. When a fraction is raised to an exponent, each part of the fraction (numerator and denominator) is raised to the exponent individually. The key steps are:

• Identify the base and exponent. Here, the base is \(\frac{57}{61}\) and the exponent is 3.
• Raise the numerator to the power of the exponent: \(57^3 = 185193\).
• Raise the denominator to the power of the exponent: \(61^3 = 227881\).
• Combine the results to form the final fraction: \(\frac{185193}{227881}\).

This method ensures that each part of the fraction is correctly handled, giving an accurate result. Remember, always perform each step carefully to avoid mistakes.