Problem 1 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(-3 x^{2} y^{3}\right)\left(5 x^{5} y^{6}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-15x^7y^9\).

Step 1: Understand the Expression

Look at the expression \((-3x^2y^3)(5x^5y^6)\). We are asked to simplify this product of two terms, each term consists of constants and variables raised to a power.

Step 2: Multiply the Coefficients

Separate the constants from the variables in each term. Multiply the constants: \(-3\) and \(5\) to get \(-15\).

Step 3: Apply the Product of Powers Rule for Variables

For variables with the same base, use the property \(a^m \, a^n = a^{m+n}\). First, apply this rule to the \(x\) terms: \(x^2 \, x^5 = x^{2+5} = x^7\). Then apply it to the \(y\) terms: \(y^3 \, y^6 = y^{3+6} = y^9\).

Step 4: Write the Simplified Expression

Combine the results of multiplying the coefficients and adding the exponents for each variable. The simplified expression is \(-15x^7y^9\).

Key Concepts

Product of Powers

When dealing with algebraic expressions, the "product of powers" property is crucial when multiplying terms with the same base. This rule states that when you multiply two powers with the same base, you can simply add the exponents together. For example, when you come across something like \(x^m \cdot x^n\), you can simplify it to \(x^{m+n}\). This concept makes calculations easier and expressions less cumbersome.

• Identify terms with the same base.
• Add the exponents of these terms together.
• Keep the base the same.

In the given expression \((-3 x^{2} y^{3})(5 x^{5} y^{6})\), we apply the product of powers to both \(x\) and \(y\) separately. For \(x\), we see \(x^2\) and \(x^5\), which becomes \(x^{2+5} = x^7\). Likewise, for \(y\), there's \(y^3\) and \(y^6\), simplifying to \(y^{3+6} = y^9\). By following these steps, you can effectively simplify any set of powers with the same base.

Simplifying Expressions

Simplifying expressions in algebra involves making them as compact as possible while ensuring they convey the same mathematical information. This often means reducing complex products or sums into their simplest forms. Here are the steps involved:

• Identify like terms or expressions that can be combined.
• Apply relevant algebraic rules such as the product of powers for simplification.
• Recalculate coefficients and simplify them if possible.

In our specific expression, we first deal with the numerical coefficients. Here, \(-3\) and \(5\) are our constants. By multiplying them, we get \(-15\). Then, we apply the concept of the product of powers to the variables \(x^2\) and \(x^5\), as well as to \(y^3\) and \(y^6\). This step simplifies the whole expression to \(-15x^7y^9\). By following these steps, any algebraic expression becomes easier to manipulate and understand.

Multiplying Monomials

Monomials are algebraic expressions containing a single term, and multiplying them involves a straightforward application of a few rules. Each monomial typically consists of a coefficient and multiple variables raised to an exponent. Here’s how to approach multiplying monomials:

• Multiply the coefficients of each monomial.
• Apply the product of powers rule to variables with the same base.
• Write the result as a single, simplified expression.

Looking at the example \((-3 x^{2} y^{3})(5 x^{5} y^{6})\), we start with multiplying the coefficients: \(-3 \cdot 5 = -15\). Next, we apply the product of powers to each set of similar bases. The \(x\)-terms multiply to become \(x^7\) and the \(y\)-terms become \(y^9\). Finally, we bring together these simplified pieces to form the complete expression \(-15x^7y^9\). This methodology simplifies not only this problem but paves the way for tackling any similar multiplication of monomials in algebra.