Problem 68 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ (x y)^{\Delta} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((xy)^\Delta\), assuming all bases are nonzero and all variable exponents are natural numbers. Answer: \((x^{\Delta})(y^{\Delta})\)

Step 1: Identify the Power Rules Needed

First, recognize that \((xy)^\Delta\) is a product of two variables raised to an exponent. To simplify this expression, we need to apply the power of a product rule, which states that \((ab)^n = a^n b^n\) for any nonzero numbers a and b, and any natural number n.

Step 2: Apply the Power of a Product Rule

Now, apply the power of a product rule to our expression \((xy)^\Delta\): $$ (x y)^{\Delta} = (x^{\Delta})(y^{\Delta}) $$ So the simplified expression is \((x^{\Delta})(y^{\Delta})\).

Key Concepts

Power of a Product Rule

The power of a product rule is an important concept when dealing with exponents in algebra. It is a handy tool that simplifies complex expressions. You might wonder, why is it useful? If you have a product raised to an exponent, each part of the product can be separately raised to that exponent. This is what the rule expresses. For example, for an expression like

• \((ab)^n\), each base \(a\) and \(b\) is raised to the power \(n\).

Thus, following the power of a product rule, it becomes \(a^n \cdot b^n\). This is especially useful when dealing with variables and makes computations more straightforward.This rule aids in breaking down and simplifying expressions, ensuring that by the time we're done, we have a much simpler form to work with.

Exponents

Exponents are a fundamental aspect of mathematics and algebra. They represent repeated multiplication of a number by itself. Essentially, if you see something like \(a^n\),

• a is called the base,
• and n is called the exponent or power.

A practical example is \(3^4\), which means 3 multiplied by itself 4 times, or \(3 \times 3 \times 3 \times 3\).Exponents have several rules, such as the power of a product rule explained earlier. Other common rules include

• the product of powers (like \(a^m \cdot a^n = a^{m+n}\)),
• and the power of a power (as in \((a^m)^n = a^{m\cdot n}\)).

These rules help simplify expressions and solve equations that involve exponentiation. Understanding exponents and their rules is crucial as they appear frequently in algebra and higher-level mathematics.

Simplification of Expressions

Simplifying expressions is a key skill in mathematics. It involves rewriting an expression in its simplest form, which makes calculations more manageable. For expressions involving exponents and products, applying rules like the power of a product rule can greatly aid in simplification.Take, for instance, the expression

• \((xy)^\Delta\),

First, identify the key components:

• "x" and "y" are bases
• and "\(\Delta\)" is the shared exponent.

Apply the power of a product rule by raising each base to the exponent separately, converting the expression to \(x^\Delta \cdot y^\Delta\).Simplification not only makes an expression easier to understand but also helps in solving equations efficiently. Fundamentally, simplification reduces complexity, allowing for more focus on solving problems rather than getting bogged down by cumbersome algebraic expressions.