Question

Multiply and reduce. Do some by calculator. $$\frac{a^{4} b^{4}}{2 a^{2} y^{n}} \cdot \frac{a^{2} x}{x y^{n}}$$

Short answer

\( \frac{a^4 b^4 a^2 x}{2 a^2 y^n x y^n} = \frac{a^6 b^4 x}{2xy^{2n}} \)

Step-by-step solution

Identify like terms to multiply

First, multiply the numerators and the denominators separately, combining like terms.

Apply exponent rules

Utilize the properties of exponents to simplify the terms. When multiplying powers with the same base, add the exponents.

Cancel common factors

Divide any common factors in the numerator and denominator to reduce the fraction to its simplest form.

Final simplification

Combine all the simplified terms to express the final answer.

Key concepts

Multiplying Polynomials

When you come across a problem involving the multiplication of polynomials, such as the provided exercise, the process involves multiplying each term in one polynomial by each term in the other. In simpler terms, it's similar to applying the distributive property extensively. Multiply the coefficients, and for any variables, just add their exponents if the bases are the same.

For example, multiplying \(a^4b^4 \times a^2x \times \frac{1}{2a^2y^n \times xy^n}\) shows the polynomial multiplication in action. Your final expression is achieved by combining like terms, which essentially means grouping and combining terms that have the same variable to the same power.

Properties of Exponents

The properties of exponents are crucial when simplifying algebraic fractions involving exponential terms. Exponent rules dictate how to handle multiplication, division, and powers of powers. One fundamental rule used here is that when multiplying powers with the same base, you add their exponents. Hence, for the term \(a^4 \times a^2\), you would add the exponents to get \(a^6\).

It’s also important to recognize that any term raised to the zero power is equal to 1, and when you divide like bases, you subtract the exponents (as you would potentially see with the variable \(y\) in our exercise). Understanding these simple but essential rules can demystify many steps towards simplifying expressions with exponents.

Reducing Fractions

Reducing fractions is all about finding common factors in the numerator and denominator and canceling them out. In the context of algebraic fractions, this involves canceling out common algebraic terms. For instance, if both the numerator and denominator have a factor of \(a^2\), they can be divided by \(a^2\), effectively canceling this factor out.

It’s similar to how \(6/8\) can be reduced to \(3/4\) by dividing both numerator and denominator by 2. Always remember, reducing an algebraic fraction to its simplest form makes it easier to work with and understand—this is your goal when simplifying algebraic expressions.

Simplifying Expressions

Finally, simplifying expressions is a sort of 'tidy up' process that combines all the earlier steps to express the original problem in its simplest form. After multiplying polynomials and applying exponent properties, you often end with terms that can be simplified further. In our exercise, once the multiplication is done, and the exponent rules are applied, you need to eliminate any common factors.

Simplifying the expression allows the final answer to be expressed in the most reduced form possible, thereby making it more comprehensible and aesthetically cleaner. It’s like cleaning up a room after a messy search—you put everything in its right place and throw out what you don't need, leaving behind an organized space.