Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(-4 b^{3}\right)\left(\frac{1}{6} b^{2}\right)\left(-9 b^{4}\right) $$

Step-by-Step Solution

Answer

6b^{9}

Step 1: Understand the Expression

We are given the expression \((-4b^{3})\left(\frac{1}{6} b^{2}\right)(-9 b^{4})\). The task is to simplify this expression by considering both coefficients and powers of \(b\).

Step 2: Simplify the Coefficients

First, simplify the numerical coefficients: \(-4\), \(\frac{1}{6}\), and \(-9\). Multiplying the coefficients together, we have: \(-4 \times \frac{1}{6} \times -9 = 6\).

Step 3: Use the Product of Powers Property

For the bases with the same variable \(b\), use the property of exponents: \(a^m \times a^n = a^{m+n}\). Here, add the exponents of \(b\). We have \(b^{3}\), \(b^{2}\), and \(b^{4}\). So, \(b^{3+2+4} = b^{9}\).

Step 4: Combine the Results

Combine the simplified coefficient with the new exponent of \(b\): This gives \(6b^{9}\).

Step 5: Final Simplified Expression

The fully simplified expression is \(6b^{9}\).

Key Concepts

Coefficients

In algebra, coefficients are the numerical parts of the terms involving variables. In the expression \(-4b^{3}\left(\frac{1}{6} b^{2}\right)(-9 b^{4})\), the coefficients are the numbers:

• -4, which multiplies the first term \(b^3\)
• \(\frac{1}{6}\), multiplying the second term \(b^2\)
• -9, which multiplies the third term \(b^4\)

To simplify expressions, you should first deal with these coefficients.
The process involves multiplying them together:

• First, multiply -4 by \(\frac{1}{6}\), giving you \(-\frac{4}{6}\) or \(-\frac{2}{3}\)
• Then, multiply that result by -9, which gives you \(6\)

After simplifying the coefficients, you'll have an easier time handling the variable parts of the expression.
Remember, coefficients are crucial as they directly influence the magnitude and direction (positive or negative) of the terms they are attached to.

Exponents

Exponents represent how many times a base, like our variable \(b\), is used as a factor. They follow specific rules that make expressions easier to handle when you simplify them.
In the expression given, the variable part involves the base \(b\) raised to different exponents:

• \(b^3\), meaning \(b \times b \times b\)
• \(b^2\), meaning \(b \times b\)
• \(b^4\), meaning \(b \times b \times b \times b\)

The main goal when simplifying expressions with exponents is to consolidate these powers into one.
This is achieved by adding the exponents together, thanks to the exponent rule that applies when multiplying like bases: \(a^m \times a^n = a^{m+n}\). This foundational concept allows for the transformation of products into a single term.
For the given expression, by adding the exponents \(3, 2,\) and \(4\), the resulting exponent is \(9\), leading us to \(b^9\). This rule greatly simplifies calculations and expressions, making it easier to interpret and use them in further math problems.

Product of Powers Property

The Product of Powers Property is a critical rule when working with expressions that include exponents. It states: when you multiply two powers with the same base, you simply add the exponents.
For instance, with our expression \((-4b^{3})\left(\frac{1}{6} b^{2}\right)(-9 b^{4})\), the base \(b\) is repeated. Thus, applying the Product of Powers Property simplifies the handling of such terms:

• Identify identical bases, which are all \(b\) in this scenario.
• Use the property \(a^m \times a^n = a^{m+n}\) to combine into a single term.
• Here, \(b^{3} \times b^{2} \times b^{4}\) becomes \(b^{3+2+4} = b^{9}\).

The beauty of this property lies in its simplicity and the efficiency it provides when simplifying lengthy expressions.
It reduces the complexity and helps keep errors at bay through a systematic approach.
Understanding and utilizing the Product of Powers Property is vital not only in algebra but also in calculus and beyond.