Problem 39 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ b^{4}\left(\frac{1}{3} b^{2}\right)\left(12 b^{-8}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4}{b^2} \).

Step 1: Combine the Terms

The expression given is \( b^4 \times \left( \frac{1}{3} b^2 \right) \times 12 b^{-8} \). Start by simplifying the constants and the powers of \( b \). Multiply the constants: \( \frac{1}{3} \times 12 = 4 \). So the expression simplifies to:\[ 4 b^4 \times b^2 \times b^{-8} \].

Step 2: Apply the Product of Powers Rule

According to the product of powers rule, when multiplying expressions with the same base, you add their exponents. Here, the base is \( b \). Add the exponents:\[ 4 + 2 + (-8) = -2 \]. Therefore, the expression becomes:\[ 4 b^{-2} \].

Step 3: Eliminate Negative Exponent

To eliminate the negative exponent, use the rule \( a^{-n} = \frac{1}{a^n} \). Here, \( b^{-2} = \frac{1}{b^2} \). Substitute, which yields:\[ 4 \cdot \frac{1}{b^2} = \frac{4}{b^2} \].

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill in mathematics that helps to make complex equations more manageable. When simplifying expressions, you aim to combine like terms and reduce the equation to its simplest form. This process often involves identifying terms that can be combined, such as constants and variables with the same base or exponent.

In our exercise, the initial expression involves a mix of coefficients and exponents with the same base, that is, the variable \( b \).

• Start by identifying and combining constants: In this problem, the constants are \( \frac{1}{3} \) and \( 12 \), which simplify to \( 4 \) when multiplied together.
• Next, organize the expression by grouping like terms: Here, it means isolating terms with the base \( b \)

Combining these steps allows for a clearer expression and sets the stage for further simplification using the property of exponents.

Negative Exponents

Dealing with negative exponents is crucial when simplifying algebraic expressions, as they can make an equation seem more complicated than it is. A negative exponent, like \( b^{-n} \), denotes the reciprocal of the base raised to the corresponding positive exponent, i.e., \( \frac{1}{b^n} \).

In our exercise, you encounter the term \( b^{-8} \). To handle this, remember:

• A negative exponent indicates how many times to divide by the base instead of multiplying by it.
• Negative exponents always suggest taking the reciprocal of the base raised to the corresponding positive power.

By converting \( b^{-8} \) to its reciprocal, you lay the groundwork to proceed with combining terms, particularly when applying the product of powers rule to simplify further.

Product of Powers Rule

The product of powers rule is an essential exponent rule used when multiplying terms with the same base. According to this rule, when you multiply terms with the same base, you add their exponents together: \( a^m \times a^n = a^{m+n} \).

In the context of our expression, this rule is applied to the powers of \( b \): \( b^4 \), \( b^2 \), and \( b^{-8} \). When combined, the exponents are added:

• Additive sequence: \( 4 + 2 + (-8) = -2 \)
• The result is \( b^{-2} \), which shows the combination of the multiply terms.

By using the product of powers rule, the expression becomes easier to work with, enabling the negative exponent to be simplified further by converting it to a positive exponent expressed as a fraction, ultimately resulting in \( \frac{4}{b^2} \).