Problem 10 - Intermediate Algebra

Question

Simplify each numerical expression. \(\left(\frac{2}{7}\right)^{-2}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{49}{4} \).

Step 1: Understand Negative Exponent

A negative exponent indicates that you need to take the reciprocal (or the inverse) of the base and change the sign of the exponent to positive. For the expression \( \left( \frac{2}{7} \right)^{-2} \), the negative exponent will transform it to \( \left( \frac{7}{2} \right)^2 \).

Step 2: Apply Positive Exponent

Now, we need to apply the positive exponent of 2 to the reciprocal fraction: \( \left( \frac{7}{2} \right)^2 \). This means multiplying \( \frac{7}{2} \) by itself: \( \frac{7}{2} \times \frac{7}{2} \).

Step 3: Multiply the Fractions

To multiply the fractions, multiply the numerators and the denominators separately: \( 7 \times 7 = 49 \) and \( 2 \times 2 = 4 \). Thus, \( \left( \frac{7}{2} \right)^2 = \frac{49}{4} \).

Key Concepts

Reciprocal

When we come across negative exponents, the key to managing them is understanding the concept of a reciprocal. A reciprocal is essentially "flipping" a fraction on its head. For example, the reciprocal of \( \frac{2}{7} \) is \( \frac{7}{2} \). Reciprocals help us turn negative exponents into positive ones, which are easier to work with.

Handling negative exponents involves taking the reciprocal of the base. This term, base, simply refers to the primary number or expression raised by an exponent. When you see a negative exponent, like \( \left( \frac{2}{7} \right)^{-2} \), convert the fraction’s base using its reciprocal. This changes the expression with a negative exponent to one with a positive exponent: \( \left( \frac{7}{2} \right)^{2} \).

This transformation sets us up for simpler arithmetic operations and helps us progress towards the expression's simplification.

Fraction Multiplication

Once you have the fraction with a positive exponent, like \( \left( \frac{7}{2} \right)^2 \), it’s time to multiply. Multiplication of fractions involves a straightforward process.

Here are the steps to multiply fractions:

• Multiply the numerators (top numbers) to get the new numerator.
• Multiply the denominators (bottom numbers) to get the new denominator.

In our example, we multiply \( \frac{7}{2} \) by itself: \( \frac{7}{2} \times \frac{7}{2} \). This results in:- Numerator: \( 7 \times 7 = 49 \)- Denominator: \( 2 \times 2 = 4 \)

You’ll find that fraction multiplication is a simple, orderly process where keeping track of multiplication across numerators and denominators is key.

Simplifying Expressions

After performing multiplication on fractions, it is crucial to check if the resulting expression can be simplified further. Simplification makes the expression clearer and easier to understand.

Once you have the result from the multiplication, check if both the numerator and the denominator share any common factors. If they do, these can be divided out to make the fraction simpler. However, in the case of \( \frac{49}{4} \), this fraction is already in its simplest form as 49 and 4 have no common factors other than 1.

Remembering these steps in simplifying expressions will help streamline your calculations and lead to more polished answers. Simplifying ensures that you are working with the simplest and most efficient form of any mathematical expression.