4/7 To The Power Of 2 As A Fraction

4/7 to the Power of 2 as a Fraction: A Comprehensive Guide

Understanding exponents, particularly when applied to fractions, is a fundamental concept in mathematics. This article delves into the detailed process of calculating (4/7)² as a fraction, explaining the underlying principles and providing practical examples to solidify your understanding. We'll also explore related concepts and applications to broaden your mathematical knowledge.

Understanding Exponents and Fractions

Before we tackle the specific problem of (4/7)², let's review the basics of exponents and fractions.

Exponents (Powers or Indices)

An exponent, also known as a power or index, indicates how many times a base number is multiplied by itself. For example, in the expression xⁿ, 'x' is the base, and 'n' is the exponent. This means 'x' is multiplied by itself 'n' times. Thus, x² = x * x, x³ = x * x * x, and so on.

Fractions

A fraction represents a part of a whole. It's expressed as a ratio of two integers, the numerator (top number) and the denominator (bottom number). For instance, in the fraction a/b, 'a' is the numerator, and 'b' is the denominator. The denominator cannot be zero.

Calculating (4/7)²

Now, let's address the core question: what is (4/7)² as a fraction?

To calculate (4/7)², we need to square both the numerator and the denominator. This means multiplying the numerator by itself and the denominator by itself.

(4/7)² = (4/7) * (4/7)

Following the rules of fraction multiplication:

• Multiply the numerators: 4 * 4 = 16
• Multiply the denominators: 7 * 7 = 49

Therefore:

(4/7)² = 16/49

This fraction, 16/49, is in its simplest form because 16 and 49 share no common factors other than 1. This means it cannot be further simplified.

Expanding the Understanding: Exponents with Negative Fractions

Let's explore a slightly more complex scenario involving negative fractions and exponents. Consider (-4/7)². The negative sign within the parentheses means we're squaring -4/7.

(-4/7)² = (-4/7) * (-4/7)

Remember that multiplying two negative numbers results in a positive number. Therefore:

• Multiply the numerators: (-4) * (-4) = 16
• Multiply the denominators: 7 * 7 = 49

Thus:

(-4/7)² = 16/49

Notice that even with a negative fraction as the base, squaring it results in a positive fraction. This is because the negative sign is also squared, effectively canceling itself out.

Exponents with Fractions: A General Rule

The process we followed for (4/7)² can be generalized for any fraction raised to a power:

(a/b)ⁿ = aⁿ / bⁿ

This means that to raise a fraction to a power, you raise both the numerator and the denominator to that power individually. This rule holds true for both positive and negative fractions.

Applications and Examples

The concept of raising fractions to powers has numerous applications in various fields:

• Geometry: Calculating areas and volumes of shapes often involves raising fractions to powers. For instance, finding the area of a square with sides of length 3/4 units involves calculating (3/4)², which is 9/16 square units.

• Physics: Many physics equations incorporate exponents and fractions, particularly those dealing with proportionality and inverse relationships.

• Finance: Compound interest calculations involve raising fractions (representing the interest rate) to powers (representing the number of compounding periods).

• Probability and Statistics: Probability calculations frequently involve raising fractions to powers, especially when dealing with independent events.

Let's look at a few more examples:

1. (2/3)³: Following the rule (a/b)ⁿ = aⁿ / bⁿ, we get (2/3)³ = 2³/3³ = 8/27

2. (-1/2)²: This is equivalent to (-1/2) * (-1/2) = 1/4. Note the negative sign disappears due to the even power.

3. (5/2)⁴: This expands to (5/2) * (5/2) * (5/2) * (5/2) = 625/16

4. (1/10)⁻²: A negative exponent indicates a reciprocal. (1/10)⁻² = (10/1)² = 100

Dealing with More Complex Exponents

While we've focused on integer exponents, it's crucial to understand how to handle fractional and decimal exponents. These scenarios require a deeper understanding of roots and logarithms which are beyond the scope of this introductory article. However, for the sake of completeness, we can introduce the idea.

A fractional exponent represents a combination of power and root. For example, (a/b)^(m/n) is equivalent to the nth root of (a/b)^m. Calculating this would require additional mathematical tools.

Simplifying Fractions

After calculating a fraction raised to a power, it's often necessary to simplify the result to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. For example, if we had 12/18, the GCD is 6. Dividing both numerator and denominator by 6 simplifies the fraction to 2/3.

Finding the GCD can be done through several methods, including prime factorization and the Euclidean algorithm. These techniques are essential for working efficiently with fractions.

Conclusion

Calculating (4/7)² as a fraction, and more broadly, understanding how to raise fractions to powers, is an essential skill in mathematics. By mastering the fundamental principles of exponents and fractions, and by practicing the techniques explained in this article, you'll be well-equipped to tackle more complex mathematical problems. Remember that the key is to apply the rule (a/b)ⁿ = aⁿ / bⁿ consistently and to simplify your answers whenever possible. Further exploration of fractional and negative exponents will significantly broaden your mathematical toolkit.