Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(8^{0}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Recognize the Zeroth Power

Any non-zero number raised to the power of zero is equal to 1. Therefore, we start by simplifying inside the parentheses: \[ 8^{0} = 1 \]

Step 2: Apply the Power of a Power Rule

Next, we use the power of a power rule which states that \((a^m)^n = a^{m imes n}\) to simplify the expression further: \[ (1^{2}) = 1^{2} \]

Step 3: Simplify the Expression

Since any number raised to any power is itself when that number is 1 (except undefined powers), further simplify: \[ 1^{2} = 1 \]

Key Concepts

Exponents

Exponents are a fundamental part of mathematics and are used to express repeated multiplication. For example, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent. This means that we multiply the base \(a\) by itself \(n\) times.

Understanding exponents allows us to work with very large or small numbers efficiently. Instead of writing out all the factors of a number, we use exponents for simplicity.

Some important points to remember about exponents include:

• When multiplying like bases, add the exponents: \(a^m \times a^n = a^{m+n}\)
• When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\)
• An exponent of one means the number itself: \(a^1 = a\)

These properties make it easier to handle mathematical operations involving powers.

Zeroth Power

The zeroth power is a unique and slightly mysterious concept in mathematics. It states that any non-zero number raised to the power of zero is equal to one. So, for instance, \(8^0 = 1\). It might initially seem confusing, but here's a simple way to think about it:

Mathematically, when you decrease the exponent by one, you're technically dividing by the base each time. If you start decreasing from \(8^1 = 8\) to \(8^0\), you divide by 8, resulting in 1. This is consistent across all numbers because it's based on the pattern of division.

This convention is helpful for keeping the consistency of mathematical operations involving powers, especially when simplifying expressions. Remember,

• \(b^0 = 1\) for any non-zero number \(b\)
• This helps in simplifying more complex algebraic expressions.

By understanding the zeroth power, you can more effectively manage expressions that might otherwise seem puzzling.

Power of a Power Rule

The power of a power rule is a fundamental exponent rule that simplifies expressions involving exponents raised to another power. According to this rule, when raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).

This rule is particularly useful for simplifying expressions, making calculations quicker and often more accessible. It helps turn complex expressions into simpler forms by following structured steps.

For example, consider \((3^2)^3\). Using the power of a power rule, we can multiply the exponents: \(3^{2\times 3} = 3^6\). This simplifies our approach and avoids extensive computations.

• The rule generally applies to any non-zero base.
• Always ensure you're multiplying the exponents, not adding or subtracting them.

By mastering this rule, you gain confidence in handling a wide range of exponent-related problems.