Problem 22 - 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. $$ (n m)^{7} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((n m)^7\). Answer: The simplified expression is $$n^7 m^7$$.

Step 1: Apply the power rule for products

The power rule for products states that \((ab)^n = a^nb^n\) for any non-zero numbers a and b and natural number n. In our case, a=n, b=m, and n=7. We can apply this rule to our expression: $$(n m)^7 = n^7 m^7$$

Step 2: Write the simplified expression

The simplified expression using the power rules for exponents is: $$n^7 m^7$$

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. When you raise a number, known as the base, to the power of an exponent, you're multiplying the base by itself exponent times. For example, in the expression \(3^4\), 3 is the base and 4 is the exponent, indicating that you need to multiply 3 by itself 4 times: \(3 \times 3 \times 3 \times 3 = 81\).

Understanding exponentiation is fundamental in algebra, especially when dealing with simplifying algebraic expressions. It simplifies large-scale multiplication and provides a way to deal with repeated multiplication of the same number in an efficient manner.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing the expression to its simplest form. This process makes expressions easier to work with and often requires the use of exponentiation. The key to simplifying expressions using exponentiation is recognizing patterns and applying the correct properties of exponents. In the example problem, breaking down \((n m)^{7}\) requires us to distribute the exponent over both n and m independently.

To simplify an algebraic expression, such as \((n m)^7\), we use the power rule for exponents to write \(n^7 m^7\), a much simpler form. This form is easier to comprehend and use in subsequent calculations or algebraic manipulations. When simplifying expressions, always ensure that you follow the hierarchy of operations and employ the correct rules for exponents to avoid common mistakes.

Properties of Exponents

Properties of exponents are the rules that govern the operation of exponentiation. They are essential tools for simplifying expressions and solving equations involving exponents. Some of the most important properties include:

• The Product Rule: \(a^n \times a^m = a^{n+m}\), which tells us that when multiplying like bases, you add the exponents.
• The Quotient Rule: \(\frac{a^n}{a^m} = a^{n-m}\), which shows that when dividing like bases, you subtract the exponents.
• The Power Rule: \((a^n)^m = a^{nm}\), which is used in our original exercise, it indicates that a power raised to another power means multiplying the exponents.
• The Zero Exponent Rule: \(a^0 = 1\), which states any number except zero raised to the exponent zero is one.
• The Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\), which relates positive and negative exponents.

By mastering these properties, students can greatly simplify complex algebraic expressions. In the context of our exercise, the power rule is appropriately applied to each factor within the parentheses, leading to the simplified expression \(n^7 m^7\).