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Chapter 10: Problem 244

In the following exercises, simplify. $$8 \cdot m^{0}-4 \cdot n^{0}$$

Short Answer

Expert verified
4

Step by step solution

01

Identify the power of zero

Recognize that both terms inside the expression have variables raised to the zero power. Recall that any non-zero number raised to the power of zero is 1.
02

Simplify each term

Simplify the terms using the rule that any number to the power of zero equals 1: \( m^{0} = 1 \) and \( n^{0} = 1 \). Hence, the expression simplifies to: \( 8 \times 1 - 4 \times 1 \).
03

Perform multiplication

Carry out the multiplication in the simplified expression: \( 8 \times 1 = 8 \) and \( 4 \times 1 = 4 \).
04

Subtract the terms

Subtract the second term from the first: \( 8 - 4 \).
05

Arrive at the simplified result

The result of the subtraction is \( 8 - 4 = 4 \). So, the simplified form of the original expression is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of Zero
When dealing with powers, the 'power of zero' can be a bit tricky at first but it's really simple once you understand it. Any non-zero number raised to the power of zero equals 1. This is a foundational rule in mathematics. For example, both \(m^0\) and \(n^0\) in the given problem become 1 because of this rule. It doesn't matter what the base (the number that's being raised to the power) is, as long as it's not zero, the result is always 1.

A quick tip: Whenever you see anything raised to zero power, just replace it with 1 right away. This will make simplifying expressions much easier.
Multiplication
Multiplication is one of the basic operations in math. In the simplification problem, multiplication comes into play after we've handled the power of zero. For example, in \(8 \times m^0 - 4 \times n^0\), since \(m^0 = 1\) and \(n^0 = 1\), it's simplified to \(8 \times 1 - 4 \times 1\).

Here are the multiplication steps: Multiplying 8 by 1 gives us 8. Similarly, multiplying 4 by 1 gives us 4. The process is straightforward because any number multiplied by 1 remains the same. This rule makes our job of simplifying expressions more manageable.
Subtraction
Subtraction is another fundamental arithmetic operation. After the multiplication step in our problem, we get \(8 - 4\).

Subtraction is simply taking one number away from another. Remember these properties:
  • Subtraction is not commutative: \(a - b eq b - a\)
  • It's also not associative: \((a - b) - c eq a - (b - c) \)
In our example, \(8 - 4\) is straightforward and equals 4. This is our final answer for the simplified expression.

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