Problem 16 - Elementary Algebra

Question

For Problems \(1-20\), find the value of each numerical expression. For example, \(2^{4}=2 \cdot 2 \cdot 2 \cdot 2=16\). $$ \left(-\frac{4}{3}\right)^{2} $$

Step-by-Step Solution

Answer

The value is \( \frac{16}{9} \).

Step 1: Understand the Base and Exponent

The given expression is \( \left(-\frac{4}{3}\right)^{2} \). This means \(-\frac{4}{3}\) is the base, and it needs to be multiplied by itself because the exponent is 2.

Step 2: Multiply the Base by Itself

The expression \( \left(-\frac{4}{3}\right)^{2} \) means \(-\frac{4}{3} \times -\frac{4}{3}\). First, calculate the multiplication of numerators: \(-4 \times -4 = 16\). Second, calculate the multiplication of denominators: \(3 \times 3 = 9\).

Step 3: Simplify the Fraction

Combine the results from multiplying the numerators and denominators to get \( \frac{16}{9} \). Therefore, the value of the expression is \( \frac{16}{9} \).

Key Concepts

Understanding Numerical Expressions

A numerical expression is a mathematical phrase involving only numbers and operation symbols. These expressions do not contain any variables. They are purely numerical and can encompass operations such as addition, subtraction, multiplication, division, or exponentiation.
For example, in the expression \((-\frac{4}{3})^2\), we are tasked with finding its value by applying the rules of exponentiation and multiplication.
Numerical expressions require careful attention to the order of operations, sometimes referred to by the mnemonic "PEMDAS" which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Understanding how to process each component step-by-step is essential in simplifying and evaluating such expressions accurately.

Working with Fractions

Fractions consist of two parts: a numerator and a denominator. The numerator is the top number, indicating how many parts of the whole are considered, while the denominator is the bottom number, indicating the total number of equal parts the whole is divided into.
In the fraction \(-\frac{4}{3}\), \(-4\) is the numerator, and \(3\) is the denominator. The negative sign indicates this is a negative fraction.
When dealing with fractions in expressions, it is crucial to handle both parts (numerator and denominator) through any operation such as multiplication or division.

• To multiply fractions, you multiply the numerators together to find the new numerator.
• Similarly, multiply the denominators together to find the new denominator.

Applying these steps in the original problem helps us manage and simplify the expression efficiently.

Multiplication of Fractions and Exponentiation

Multiplication of fractions can seem tricky, but following a few straightforward steps can demystify the process. In the given expression \((-\frac{4}{3})^2\), we are asked to multiply the fraction \(-\frac{4}{3}\) by itself.
Exponentiation signifies repeated multiplication. In our case, here's how it works:

• Repeat the fraction as many times as the exponent indicates. Here, the exponent \(2\) tells us to multiply the fraction by itself once: \(-\frac{4}{3} \times -\frac{4}{3}\).
• Multiply the numerators: \((-4)\times(-4)\) which equals \(16\).
• Multiply the denominators: \(3 \times 3\) which equals \(9\).

The product \(\frac{16}{9}\) is the simplified result.
Remember, multiplying two negative numbers results in a positive, explaining why the final result \(\frac{16}{9}\) is positive.