Problem 21 - Algebra A Combined Function

Question

Evaluate \(\left(-\frac{3}{4}\right)^{2}\)

Step-by-Step Solution

Answer

The value of \((-\frac{3}{4})^2\) is \(\frac{9}{16}\).

Step 1: Identify the Base and Exponent

The expression \((-\frac{3}{4})^2\) consists of a base and an exponent. The base is \(-\frac{3}{4}\) and the exponent is 2. This means we will multiply \(-\frac{3}{4}\) by itself.

Step 2: Apply the Exponent to the Base

The expression indicates that the base \(-\frac{3}{4}\) should be multiplied by itself once (since the exponent is 2). This can be written as: \(-\frac{3}{4} \times -\frac{3}{4}\).

Step 3: Multiply the Numerators

First, multiply the numerators of the fractions: \(-3 \times -3 = 9\).

Step 4: Multiply the Denominators

Next, multiply the denominators of the fractions: \(4 \times 4 = 16\).

Step 5: Combine the Results to form the Fraction

Combine the numerator and the denominator to form the fraction: \(\frac{9}{16}\).

Key Concepts

Base and Exponent

In mathematics, expressions involving exponents have two primary parts: the base and the exponent. The base is the number or term being multiplied, while the exponent indicates how many times the base will be multiplied by itself. This is often written as \(a^n\), where \(a\) is the base and \(n\) is the exponent.
In the expression \((-\frac{3}{4})^2\), the base is \(-\frac{3}{4}\), and the exponent is 2.

• The base, \(-\frac{3}{4}\), tells us the number we are multiplying.
• The exponent, 2, tells us to multiply the base by itself once.

Recognizing the base and exponent in an expression is the first step towards solving it. It helps to clearly identify what operations must be performed.

Fraction Multiplication

Multiplying fractions involves a straightforward process. You multiply the numerators (the top numbers) to get the new numerator and multiply the denominators (the bottom numbers) to get the new denominator.

• Given fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), the product is \(\frac{a \times c}{b \times d}\).

In our example, we multiply \(-\frac{3}{4}\) by itself:
\(-\frac{3}{4} \times -\frac{3}{4}\).

• Multiply the numerators: \(-3 \times -3 = 9\).
• Multiply the denominators: \(4 \times 4 = 16\).

So, the result of the fraction multiplication is \(\frac{9}{16}\).
Fraction multiplication is essential in simplifying expressions with fractional bases and ensures accuracy in calculations.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (\(-\)). When dealing with negative numbers in multiplication, a few rules apply:

• When two negative numbers are multiplied, the result is positive.
• When a negative number is multiplied by a positive number, the result is negative.

In our problem, both factors \(-3\) are negative:
\(-3 \times -3\) resulting in a positive product of 9.
Understanding these rules about negative numbers helps ensure that you can solve problems accurately without confusion about the sign of the result.

Squared Terms

A squared term refers to a number or expression raised to the power of 2. This is noted by the exponent 2 and is a critical concept in algebra.

• Squaring a number means multiplying it by itself.
• For example, if \(x\) is squared, it's expressed as \(x^2 = x \times x\).

In the exercise \((-\frac{3}{4})^2\), square the base \(-\frac{3}{4}\) to get:
\(-\frac{3}{4} \times -\frac{3}{4} = \frac{9}{16}\).
This operation confirms the result when we square negative fractions. Squared terms are fundamental in understanding quadratics, area of squares, and more.