Question

Write with a single exponent. $$ \left((x+y)^{4}\right)^{5} $$

Short answer

Question: Simplify the expression \({(x+y)^4}^5\). Answer: \((x+y)^{20}\)

Step-by-step solution

Identify the base and the two exponents

In our expression, the base is \((x+y)\), and the two exponents are \(4\) and \(5\).

Apply the power to a power rule

According to the power to a power rule, we need to multiply the exponents: \({(a^m)^n = a^{mn}}\). In this case, \({(x+y)^4}^5 = (x+y)^{4\cdot5}\).

Calculate the new exponent

Multiply the exponents: \(4\cdot5 = 20\). Now we can rewrite the expression with a single exponent: \((x+y)^{4\cdot5}=(x+y)^{20}\).

Write the final simplified expression

The simplified expression with a single exponent is \((x+y)^{20}\).

Key concepts

Base and Exponents

In mathematics, the terms 'base' and 'exponents' are fundamental concepts used in expressing powerful techniques of multiplication in a compact form. The base is the number or expression that is being multiplied. The exponent tells us how many times we multiply the base by itself. In our original exercise, the expression \((x+y)^4\) has \((x+y)\) as its base, and the number 4 is the exponent. Here are some more key points to keep in mind:

• The base can be any number, variable, or more complex expression like \((x+y)\).
• An exponent is also known as a power, and it indicates repeated multiplication.

For example, \(2^3\) means 2 multiplied by itself three times: \(2 \times 2 \times 2\). Moving from simple numbers to algebraic expressions like \((x+y)^4\), the concept remains the same, where \((x+y)\) is multiplied four times by itself. It's crucial to identify the base and exponent correctly as this forms the foundation for applying various exponent rules.

Power to a Power Rule

The power to a power rule is an exponent rule that simplifies expressions where an exponent is raised to another exponent. This rule states that when you have a power raised to another power, you multiply the exponents. Mathematically, this is represented as \((a^m)^n = a^{m \cdot n}\). This is a key rule used to simplify more complex expressions involving exponents.
Let's break it down:

• Identify the inner exponent first. In \((x+y)^4\), the 4 is the inner exponent.
• Then, look at how this expression is raised to an additional exponent, 5, forming \(((x+y)^4)^5\).
• According to the power to a power rule, multiply these exponents: \(4 \times 5 = 20\).

So, by applying this rule to our exercise, we transform \(((x+y)^4)^5\) into \((x+y)^{20}\). This simplification is crucial in mathematical expressions and calculations.

Simplified Expression

A simplified expression is one that is reduced to its most compact or easily understood form. Simplifying often involves using a set of rules and techniques to make the expression easier to work with or analyze. In our exercise, reducing \(((x+y)^4)^5\) into \((x+y)^{20}\) is simplifying the expression.

• Simplification benefits include easier computation, clearer interpretation, and reduced complexity.
• An expression is considered simplified when you can't reduce or break it down any further using the rules of mathematics.
• In this case, by applying the power to a power rule, we reduced two levels of exponents into one, achieving a streamlined form.

Always aim for the simplest expression when working with mathematical problems, as it minimizes errors and enhances understanding. Simplified expressions are especially valuable for algebraic manipulation and solving equations efficiently.