Problem 36 - 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. $$ \left(2 a^{5} b^{11}\right)^{0} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is 1.

Step 1: Recall the power rule for exponents

Remember that for any nonzero number A, \(A^{0} = 1\). This rule applies not just to numbers but also to variables with exponents and any combination of them.

Step 2: Apply the rule to the given expression

The given expression is \((2 a^{5} b^{11})^{0}\), and we can apply the power rule for exponents to this expression. Since the whole expression is raised to the power of 0, the result will be 1.

Step 3: Write the simplified expression

After applying the power rule for exponents, the simplified expression is: $$1$$

Key Concepts

Exponential Expressions

An exponential expression is a mathematical notation that indicates the operation of exponentiation, which involves raising a base number to a certain power. This power, also known as an exponent, denotes how many times the base number is multiplied by itself. For instance, in the expression \(5^3\), 5 is the base and 3 is the exponent, which means that 5 is multiplied by itself two additional times: \(5 \times 5 \times 5 = 125\).

When dealing with exponential expressions, it's essential to remember that every component, including numbers, variables, and products of these, can have its own exponent. Furthermore, there are rules and properties of exponents—such as the power rule, product rule, quotient rule, and power of a power rule—which help us simplify complex expressions efficiently. Knowing these rules and understanding how to apply them is fundamental in solving algebraic problems.

Simplifying Algebraic Expressions

To simplify an algebraic expression means to rewrite it in the most basic form while retaining its value. Simplification often involves combining like terms, expanding products, and utilizing the rules of exponents, along with other algebraic principles. An essential aspect of simplification is making the expression as concise and as straightforward as possible, which is especially helpful for evaluating the expression or solving equations in which it's included.

Simplifying expressions is not merely about making them look simple, but also about preparing them for further mathematical processes such as solving or graphing. For example, an expression like \((3x^2)^4\) can be greatly simplified by applying the power of a power rule, ending up as \(3^4x^8\), which is easier to work with in equations or graphs.

Exponentiation

Exponentiation is a mathematical operation, involving two numbers, the base \(b\) and the exponent \(n\), in the expression \(b^n\). The exponent dictates how many times the base is used as a factor in a product. For positive integers as exponents, the process is straightforward multiplication. For instance, \(2^3 = 2 \times 2 \times 2 = 8\). However, exponentiation can sometimes involve special rules.

One special exponent is zero. Any nonzero base raised to the power of zero is equal to 1, denoted as \(b^0 = 1\), where b is a nonzero number or algebraic expression. This rule is vital for simplifying expressions and is an excellent example of making complex operations manageable. Understanding that any term or expression raised to the power of zero results in one helps to simplify problems that can initially seem daunting, reducing expressions to their most straightforward form.