Problem 82 - Elementary Algebra

Question

Simplify \(\left(5 x^{2} y^{4}\right)\left(2 x y^{5}\right)\)

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(\left(5 x^{2} y^{4}\right)\left(2 x y^{5}\right)\) Answer: \(10x^3y^9\)

Step 1: Rewrite the given expression

The given expression that we need to simplify is \(\left(5 x^{2} y^{4}\right)\left(2 x y^{5}\right)\).

Step 2: Use the distributive property

In this step, we need to multiply the coefficients and the variables of the two expressions: $$(5 x^{2} y^{4})(2 x y^{5}) = 5 \cdot 2 \cdot x^2 \cdot x \cdot y^4 \cdot y^5$$

Step 3: Multiply the coefficients

Multiply the two coefficients together: $$(5 \cdot 2) = 10$$ The expression now becomes $$10 \cdot x^2 \cdot x \cdot y^4 \cdot y^5$$

Step 4: Use the rules of exponents for the x variable

To multiply the \(x\) variables, we will use the rule: \(x^m \cdot x^n = x^{m+n}\). In this case, we have \(x^2 \cdot x = x^{2+1} = x^3\). The expression now becomes $$10 \cdot x^3 \cdot y^4 \cdot y^5$$

Step 5: Use the rules of exponents for the y variable

Repeat the previous step with the \(y\) variables: \( y^4 \cdot y^5 = y^{4+5}=y^9\). The expression now becomes $$10 \cdot x^3 \cdot y^9$$

Step 6: Final simplified expression

After performing all the necessary operations, our final simplified expression is: $$10x^3y^9$$

Key Concepts

Distributive Property

The distributive property helps to simplify expressions where two groups need to be multiplied. This principle is crucial when dealing with terms inside parentheses. For expression simplification, it dictates that every term in the first group should multiply every term in the second group.
If you have an expression like \(a(b+c)\), you apply the distributive property by distributing the first term inside the parentheses, leading to \(ab + ac\).
In the case of \((5x^2y^4)(2xy^5)\), this property assists in breaking down the multiplication as \(5 \times 2 \times x^2 \times x \times y^4 \times y^5\). Each part of one expression interacts with its counterpart in the other to form a simplified result.

Rules of Exponents

Exponent rules are helpful shortcuts to simplify expressions that involve powers of the same base. Key exponents rules include:

• Product of Powers: \(x^m \cdot x^n = x^{m+n}\). This means if you multiply like bases, you add their exponents.
• Power of a Power: \((x^m)^n = x^{m \cdot n}\). Here, raising a power to another power means multiplying their exponents.
• Power of a Product: \((xy)^n = x^n \cdot y^n\). Distributes the exponent to each factor in the product.

For instance, applying the product of powers to \(x^2 \cdot x\) would simplify to \(x^{2+1} = x^3\). Similarly, \(y^4 \cdot y^5\) becomes \(y^{4+5} = y^9\).
Understanding these rules smoothens the process of polynomial simplification.

Monomial Multiplication

Monomial multiplication involves multiplying single-term algebraic expressions, which can comprise coefficients and variables raised to powers. The process incorporates both the distributive property and the rules of exponents.
With monomials, you address each part separately:

• Coefficients: Multiply these directly (e.g., \(5 \cdot 2 = 10\)).
• Variables: Apply the rules of exponents by maintaining the base and adding the exponents (e.g., \(x^2 \cdot x = x^3\)).

In the task, simplifying \( (5x^2y^4)(2xy^5) \) involves combining coefficients, then handling each variable set with exponent rules. This technique results in structured steps leading to the final expression \(10x^3y^9\). Understanding this method simplifies the complexities of polynomial multiplication.