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Chapter 9: Problem 29

Simplify. $$\left(-a^{2} b^{3}\right)\left(-a b^{2} c^{4}\right)$$

Short Answer

Expert verified
The simplified form of the expression is \( a^{3} b^{5} c^{4} \)

Step by step solution

01

Expand the expressions

Firstly, expand each term inside the given product. So \( (-a^{2} b^{3})(-a b^{2} c^{4}) \) becomes \( -a^{2} \cdot -a \cdot b^{3} \cdot b^{2} \cdot c^{4} \).
02

Use the multiplication rule of exponents

Now utilize the multiplication property of exponents which states that, for any integer n and m, a^n * a^m = a^(n+m). Thus, the expression becomes \( a^{2+1} \cdot b^{3+2} \cdot c^{4} \), which simplifies to \( a^{3} \cdot b^{5} \cdot c^{4} \).
03

Apply properties of negative numbers

Observe that each term in the given expression has a negative sign. Recall that the product of two negative numbers is positive. Therefore, in this case, the entire expression will have a positive sign.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponents
Exponents are a way to represent repeated multiplication. When you see a number like \( a^2 \), it means you multiply \( a \) by itself: \( a \times a \). The same applies to variables. If you have \( b^3 \), this means \( b \times b \times b \).

In algebra, exponents simplify complex multiplications. They follow certain rules, like the multiplication property, which we'll explain next. This makes working with repeated multiplications more efficient and manageable.
Multiplication Property of Exponents
The multiplication property of exponents is a handy tool. It states that when you multiply like bases, you can add their exponents: \( a^n \times a^m = a^{n+m} \). This property only works if the bases are the same.

So, when simplifying expressions like \( a^2 \times a \), you would add the exponents: \( 2 + 1 = 3 \). This means \( a^2 \times a = a^3 \). This rule helps to simplify expressions quickly and easily by reducing the number of terms.
Working with Negative Numbers
Handling negative numbers requires remembering a few simple rules. When you multiply two negative numbers, the result is positive. Think of it as two negatives canceling each other out.

For example, in the expression \(-a^2 \times -a\), both terms are negative. Multiplying them together will yield a positive result. It's important to apply this rule of negatives to ensure your answer is correctly simplified with the right sign.
Simplification
Simplification is all about making expressions easier to work with. By following pre-algebra rules, we can transform complex expressions into simpler forms.

For instance, in the given expression \((-a^2 b^3)(-a b^2 c^4)\), simplifying involves combining like terms and applying basic properties like those of exponents and negative numbers. By breaking down each part systematically and applying known properties, you get a clean, simplified expression such as \(a^3 b^5 c^4\). This final expression is simple, readable, and much easier to handle in further calculations.

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