Question

Challenge Problems $$\left(1.84 w x^{2} y\right)\left(2.44 w^{2} x y^{3}\right)\left(1.65 w x^{3} y\right)\left(2.33 w^{2} x y\right)$$

Short answer

The simplified expression is approximately 18.91428 w^6 x^7 y^6.

Step-by-step solution

Multiply the coefficients

First, multiply the numerical coefficients of the terms: 1.84 * 2.44 * 1.65 * 2.33.

Combine the w terms

Next, multiply the variables with the same base, starting with 'w'. Combine the exponents according to the rule of exponents for multiplication, which states that when multiplying like bases, you add the exponents. Thus, you have w^1 * w^2 * w^1 * w^2.

Combine the x terms

Then, multiply the variables with the base 'x' using the same exponent rule as before: x^2 * x * x^3 * x.

Combine the y terms

After that, multiply the variables with the base 'y': y * y^3 * y * y.

Calculate the final expression

Multiply the numerical coefficient obtained from Step 1, with the combined exponents from the w, x, and y terms.

Key concepts

Exponent Rules

Understanding exponent rules is crucial when multiplying polynomials with variables raised to different powers. Let's explore the exponent rules using the given exercise as our reference point.

The fundamental rule applied here is the Product of Powers rule, which states that when you multiply two exponents with the same base, you keep the base and add the exponents: for any nonzero base 'a' and any exponents 'm' and 'n', the rule is expressed as \( a^m \times a^n = a^{m+n} \). In our exercise, this means you combine the 'w' terms by adding their exponents: \( w^1 \times w^2 \times w^1 \times w^2 = w^{1+2+1+2} \), resulting in \( w^6 \). You repeat this process for each set of like bases ('x' and 'y' in this case) to fully simplify the expression.

Multiplying Coefficients

When multiplying polynomials, an essential step is to multiply the coefficients—the numerical part of the terms. This is straightforward arithmetic multiplication. In our example, we multiply the coefficients of each term first: \(1.84 \times 2.44 \times 1.65 \times 2.33\).

It's important to carry out this multiplication accurately to get the correct numerical coefficient for the final expression. In practice, it often involves dealing with decimals or fractions, so it might be helpful to use a calculator to ensure precision. The product of these coefficients becomes the numerical part of the final expression of the multiplied polynomial.

Variable Term Multiplication

After handling the coefficients, the focus shifts to variable term multiplication. This involves variables with exponents, and here's where our exponent rule knowledge is applied. Each variable is treated separately. For instance, with 'x' variables, you multiply each term while adhering to the same base rule: combine their exponents.

Similarly, for 'y' variables, the expression \( y \times y^3 \times y \times y \) translates to \( y^{1+3+1+1} = y^6 \). In essence, you are not multiplying the exponents; you are adding them up to find the new power for the base variable in the final polynomial expression. Combining variable terms properly ensures your final answer encapsulates all the variables correctly raised to their appropriate powers, resulting in a neat and accurate polynomial product.