Problem 61 - Elementary Algebra Within Reach

Question

In Exercises \(59-62\), rewrite the expression as a product. $$ \left(-\frac{3}{4}\right)^{2} $$

Step-by-Step Solution

Answer

The product is \(\frac{9}{16}\)

Step 1: Understand the Exponent

An exponent refers to the number of times a number is multiplied by itself. So, \(-\frac{3}{4}^2\) means \(-\frac{3}{4}\) is multiplied by itself.

Step 2: Apply the Power of Product Rule

The power of a product rule states that the power of a product of two numbers is equal to the product of each number raised to the power. In this case, \(-\frac{3}{4}^2\) can be rewritten as \((-3)^2 * (\frac{1}{4})^2\).

Step 3: Simplify the Expression

Now, we can simply calculate \((-3)^2 = 9\) and \((\frac{1}{4})^2 = \frac{1}{16}\)

Step 4: Multiply the Simplified Results

After simplifying, we multiply the results 9 and \frac{1}{16} to get \(\frac{9}{16}\)

Key Concepts

Power of a Product Rule

When dealing with exponents, understanding the power of a product rule is crucial. This rule states that when you have a product raised to an exponent, you can apply the exponent to each factor in the product separately. For instance, in the expression \((-\frac{3}{4})^2\), you can use the power of a product rule to rewrite it as \(( -3 )^2 \cdot ( \frac{1}{4} )^2\).

This approach simplifies calculations by allowing you to deal with each factor independently, making the process more straightforward.

• First, apply the exponent to \(-3\) to get \((-3)^2\).
• Next, apply the exponent to \(\frac{1}{4}\) to get \((\frac{1}{4})^2\).

These individual calculations can then be multiplied together, which often results in easier arithmetic. The power of a product rule is a powerful tool for breaking down more complex expressions into manageable pieces.

Simplifying Expressions

Simplifying expressions with exponents involves breaking down complex expressions into simpler, more easily calculable parts. Using our example, \((-\frac{3}{4})^2\), the expression can be simplified in stages.

Begin by understanding what each part of the expression does. The exponent \(^2\) tells us to multiply \(-\frac{3}{4}\) by itself. By applying the power of a product rule, we handle each element separately as \((-3)^2\) and \((\frac{1}{4})^2\).

• \((-3)^2\): Multiply \(-3\) by itself to get \(9\).
• \((\frac{1}{4})^2\): Multiply \(\frac{1}{4}\) by itself to yield \(\frac{1}{16}\).

Simplifying these elements individually makes it easy to see the expression's final form. After calculating, simply multiply the results \(9\) and \(\frac{1}{16}\) together to achieve \(\frac{9}{16}\). This method breaks down the problem into smaller parts, making the process smoother.

Algebraic Multiplication

Algebraic multiplication, particularly in the context of exponents, involves multiplying together numbers, fractions, or variables while considering their powers. For the expression \((-\frac{3}{4})^2\), the multiplication involves a product of two separate results.

Initially, using the power of a product rule, the expression is split into two parts: \((-3)^2\) and \((\frac{1}{4})^2\). After simplifying each separately, you end up with \(9\) and \(\frac{1}{16}\).

• Multiply \(9\) by \(\frac{1}{16}\): This is straightforward as it involves multiplying a whole number by a fraction.
• The result is \(\frac{9}{16}\), which represents the product of the simplified parts of the original expression.

Algebraic multiplication here shows how breaking up expressions and focusing on individual components simplifies the process. It's a practical approach to deal with more complicated algebraic expressions involving powers.