Question

Write each expression as a product or a quotient. Assume all variables are positive. $$ (r-s)^{t+z} $$

Short answer

Question: Rewrite the following expression as a product or a quotient, assuming all variables are positive: $(r-s)^{t+z}$. Answer: $(r-s)^{t+z} = (r-s)^t(r-s)^z$

Step-by-step solution

Apply the exponent rule \((a^m)^n = a^{mn}\)

In this case, our exponent is \(t+z\). According to the exponent rule, we can rewrite the expression as: $$ ((r-s)^t)((r-s)^z) $$

Rewrite the expression as a product or a quotient

The expression is now in product form as required: $$ (r-s)^{t+z} = (r-s)^t(r-s)^z $$

Key concepts

Exponent Rules

Exponent rules are essential for simplifying expressions that involve powers. They help us manage the multiplication and division of numbers with exponents easily and accurately. One useful rule states that when you multiply like bases, you can add their exponents: if you have \(a^m \times a^n\), it simplifies to \(a^{m+n}\). This is known as the Product of Powers Property.
Another important rule is the Power of a Product, which states that \((ab)^n = a^n \times b^n\). It allows distributing the external exponent to all components inside the brackets.
A similar principle applies to exponents applied over an exponent, as given by \((a^m)^n = a^{mn}\). This is known as the Power of a Power property. For instance, the expression \((r-s)^{t+z}\) can be rewritten using this rule to break it down into a simpler form by separating the bases: \(((r-s)^t)((r-s)^z)\).
Understanding these properties allows us to handle complex expressions better by repeatedly applying these rules to simplify them into a form that's easier to evaluate.

Product of Expressions

A product of expressions occurs when you multiply two or more expressions together. In algebra, reorganizing an expression as a product can often simplify it and reveal underlying patterns or constants that weren't initially obvious.
When we rewrote \((r-s)^{t+z}\) into \((r-s)^t \times (r-s)^z\), we effectively expressed it as a product of like bases, using the rule for multiplying powers with the same base. This approach makes such expressions much easier to manage.
Think of the process like breaking down a complex task into smaller, manageable ones:

• You start by identifying the common base in the expression.
• Then apply the rules of exponents to rearrange it into a product (or multiplication) of simpler expressions.
• This setup often allows for further simplifications or solutions, especially when substituting known values.

By approaching it this way, we not only simplify our calculations but gain flexibility in handling more complex problems in algebra.

Quotient of Expressions

The quotient of expressions is when one expression is divided by another. This concept is similar to handling fractions but with algebraic expressions. The basic rule here, the Quotient of Powers, says \(\frac{a^m}{a^n} = a^{m-n}\), assuming \(aeq 0\). It involves subtracting the exponents when dividing like bases.
Even though our initial task didn't focus directly on quotients, understanding division in algebraic expressions is fundamental:

• When simplifying expressions, converting multiplication into division and vice versa can be very useful.
• Much like breaking products into sums, rewriting or simplifying fractions can sometimes make an expression much clearer.
• This might involve factoring out common terms or rearranging the expression to suit the problem's needs.

Mastering both product and quotient rules provides a balanced approach to managing expressions in algebra, equipping you to tackle a broader range of algebraic problems efficiently.