Problem 71 - Algebra 1

Question

Evaluate the expression. $$\left(\frac{2}{5}\right)^{2}$$

Step-by-Step Solution

Answer

The evaluation of the expression \( (\frac{2}{5})^{2} \) is \(\frac{4}{25}\).

Step 1: Identify the components of the expression

This is a power operation applied on a fraction, where the base is \(\frac{2}{5}\) and the exponent is 2.

Step 2: Determine the power of the numerator and the denominator

To calculate the power of a fraction, take the power of the numerator and the denominator separately. So, \((\frac{2}{5})^{2}\) will be equal to \((2^{2})/(5^{2})\).

Step 3: Compute the exponent operation

Calculate the values of \(2^{2}\) and \(5^{2}\) to get 4 and 25 respectively.

Step 4: Write the final answer

The result is the fraction \(\frac{4}{25}\).

Key Concepts

Fraction Exponentiation

Fraction exponentiation is a way of applying powers to fractions, which are simply numbers written in the form of one number divided by another. When you see a fractional base like \( \left( \frac{2}{5} \right)^2 \), it means you need to raise both the numerator (top number) and the denominator (bottom number) to the power of the exponent.
This fractional base of \( \frac{2}{5} \) with an exponent of 2 means you must calculate \( 2^2 \) and \( 5^2 \) separately. This keeps things organized and straightforward, allowing you to see each part's contribution to the overall answer.
Let's highlight how easy this is:

• Raise the numerator 2 to the power of 2, which equals 4.
• Raise the denominator 5 to the power of 2, which equals 25.

This gives you a clean fractional answer of \( \frac{4}{25} \), making fraction exponentiation an approachable process!

Exponentiation Rules

Understanding exponentiation rules can unravel a lot of confusion around powers and make math more predictable. When working with exponents, especially in algebra, a few key rules guide our operations:

• Any number raised to the power of 1 remains as itself; \( a^1 = a \).
• Any number raised to the power of 0 equals 1; \( a^0 = 1 \), given \( a eq 0 \).
• The product of powers rule states that if you multiply powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• The power of a power rule tells us to multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• The power of a product rule involves distributing the exponent to each term: \( (ab)^n = a^n \times b^n \).

These rules come into play for computations like \( \left( \frac{2}{5} \right)^2 \), reminding us that we distribute the power to both the numerator and the denominator. Simplifying such expressions becomes a structured approach when these rules are at hand.

Numerator and Denominator Powers

In fraction exponentiation, knowing how to handle the powers of both the numerator and the denominator is crucial. Each is treated as a separate entity while applying the exponent, which is common in algebra.
For a fraction like \( \frac{2}{5} \) raised to the power of 2, calculate the power of the numerator: \( 2^2 = 4 \). Simultaneously compute the power of the denominator: \( 5^2 = 25 \).
By separately addressing the numerator and the denominator, you ensure each is accurately calculated, leading to the correct fractional result of \( \frac{4}{25} \). This separation of operations is the key:

• It simplifies the process by dealing with smaller, manageable numbers.
• Ensures precision by isolating each part of the fraction.
• Helps clearly visualize the transformation of each component.

Thus, focusing on the numerator and denominator powers helps in cleanly solving fractional exponentiation problems.