Problem 13 - College Algebra

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 expression simplifies to 1.

Step 1: Understanding the Exponent Zero Rule

The first thing to recognize is the rule that any non-zero number raised to the power of zero is 1. Hence, calculate the value of the base with the power. Here, it's given as \(8^0\). Therefore, \(8^0 = 1\).

Step 2: Applying the Exponentiation Rule

After calculating the expression within the parentheses, substitute the simplified expression into the outer exponent: \((8^0)^2 = (1)^2\).

Step 3: Simplifying the Expression

Now simplify \((1)^2\). The exponent 2 means multiplying 1 by itself, which gives \(1 \times 1 = 1\). Thus, \((1)^2 = 1\).

Step 4: Expressing with Positive Exponents

Since the simplified expression \(1\) already involves positive exponents, no further changes are needed. The answer is \(1\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like solving a puzzle. The aim is to make them as simple as possible, without changing their value. In our exercise, we're given \( (8^0)^2 \).

To simplify it, let's start from the inside out. First, simplify the inner expression using the Exponent Zero Rule. This means recognizing patterns and rules that can help us simplify step-by-step. Think of it as turning a complex sentence into a simpler one while keeping the meaning intact.

• Identify any exponents and apply the rules related to them.
• Simplify exponentiations step-by-step based on these rules.
• Always aim to express the answer with positive exponents if possible.

For \( (8^0) \), after using the rules, we replace this entire part with \(1\), making our expression \((1)^2\).

The final simplification of \((1)^2\) is straightforward because \(1\) times itself is still \(1\). Therefore, the expression simplifies entirely to \(1\). By focusing on simplifying expressions, you can tackle more difficult problems by breaking them down into easier parts.

Exponent Zero Rule

The Exponent Zero Rule is a fundamental concept in algebra. It states that any non-zero number raised to the power of zero equals \(1\).

To understand why, we need to remember that exponents represent how many times to multiply a number by itself. However, when the exponent is zero, there's nothing to multiply! The rule is important because it helps simplify expressions and solve problems quickly.

Here are some highlights of the Exponent Zero Rule:

• Applies to any non-zero number \(x\), so \(x^0 = 1\).
• Helps turn complicated expressions into simpler forms.
• Vital when dealing with equations and simplifications like \(8^0\), which simplifies to \(1\).

When we applied this rule to \(8^0\) in our example, we got \(1\). This rule is like a magic shortcut in math, making our calculations both quicker and clearer.

Positive Exponents

Positive exponents are crucial in mathematics. They indicate how many times a number should be multiplied by itself.

In our exercise example \((8^0)^2\), the positive exponent \(2\) shows that we should multiply the base (simplified to \(1\)) by itself that many times.

• A positive exponent like \(n\) in \(x^n\) means \(x\) is multiplied by itself \(n\) times.
• They are straightforward to work with and result in whole numbers or fractions if the base is divided rather than multiplied.
• Unlike negative or zero exponents, there's no conversion, as they already directly express the multiplication or division required.

In calculations, positive exponents simplify well and ensure our answers remain in a standard format. With \((1)^2 = 1\), the positive exponent doesn't change the value, but it's crucial to understanding how multiplication was executed. In practical terms, positive exponents make math logical and predictable.