Problem 49 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ b^{4}\left(3 a b^{3}\right)\left(2 a^{2} b^{-5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(6a^3b^2\).

Step 1: Distribute the exponents

First, calculate the product inside the parentheses by distributing the exponents over the coefficients and variables. Start by multiplying the constants and then the variables separately.

Step 2: Multiply the coefficients

Multiply the numerical coefficients: \(3\) and \(2\). So, \(3 \times 2 = 6\).

Step 3: Multiply powers of \(a\)

Combine the powers of \(a\) using the rule \(a^m \times a^n = a^{m+n}\). We have \(a\) to the power of \(1\) and \(a^2\), giving us \(a^{1+2} = a^3\).

Step 4: Multiply powers of \(b\)

Combine the powers of \(b\) using the rule \(b^m \times b^n = b^{m+n}\). Here, it's \(b^4 \times b^3 \times b^{-5}\). Add the exponents: \(4 + 3 - 5 = 2\). So, it becomes \(b^2\).

Step 5: Write the simplified expression

Combine all the multiplied parts. The final expression is \(6a^3b^2\), with no negative exponents.

Key Concepts

The Distributive Property

The distributive property is a key concept when simplifying expressions, especially those involving multiplication. Here, it is used to "distribute" the multiplication over several terms inside an expression. In our exercise, we had the expression \( b^4(3ab^3)(2a^2b^{-5}) \). We start by addressing each part separately:

• First, handle constants: multiply \(3\) by \(2\).
• Then, apply the exponents to each variable separately, multiplying them together where necessary.

Doing this ensures that all terms within the parentheses contribute systematically to the final expression. This technique helps simplify complex expressions by breaking them down into manageable parts.

Multiplying Exponents

Multiplying exponents involves the use of specific rules that simplify mathematical expressions efficiently. When you multiply variables with exponents, such as \(a^m\) and \(a^n\), you keep the base the same and add up the exponents. The general rule is:

• \(a^m \times a^n = a^{m+n}\)

In our example, we had to multiply \(a\) terms

• \(a^1 \times a^2 = a^{1+2} = a^3\).

Similarly, this principle also applies to other bases and exponents. It's a fundamental part of simplifying algebraic expressions efficiently. The power of this rule lies in its simplicity, converting a multiplication problem into a straightforward addition of exponents.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves reducing expressions to their simplest form. This often includes many steps such as applying distribution, consolidating like terms, and reducing expressions with exponents.

• Each operation within an expression plays a crucial role in the simplification process.
• The sequence of operations is vital; in our exercise, multiplying coefficients followed by applying exponent rules helps simplify the expression correctly.

Ultimately, simplifying makes the expression easier to read and interpret, eliminating any unnecessary complexity while preserving its original value. In our case, the expression reduced from a complicated combination of terms to a concise \(6a^3b^2\).

Negative Exponents

Negative exponents can be confusing, but they are just another way to express fractions. A negative exponent, like \(b^{-5}\), indicates that it's actually the reciprocal of the base raised to the positive of that exponent:

• \(b^{-5} = \frac{1}{b^5}\)

In our exercise, when simplifying, we converted our terms so there were no negative exponents left. Starting with \(b^4 \times b^3 \times b^{-5}\), combining these by adding their exponents, we got \(b^{4+3-5} = b^2\).

• This operation reads as: combine and simplify to eliminate the negative exponent.

Understanding how to work with negative exponents allows you to simplify expressions while following the right mathematical rules and principles.