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Chapter 10: Problem 243

In the following exercises, simplify. $$2 \cdot x^{0}+5 \cdot y^{0}$$

Short Answer

Expert verified
7

Step by step solution

01

Identify the Exponents

In the expression, observe the terms with exponents: \( x^{0} \) and \( y^{0} \).
02

Apply the Zero Exponent Rule

Recall that any number (except zero) raised to the power of zero is 1. Thus, \( x^{0} = 1 \) and \( y^{0} = 1 \).
03

Substitute the Values

Replace \( x^{0} \) and \( y^{0} \) with 1 in the expression. This results in the equation: \( 2 \times 1 + 5 \times 1 \).
04

Simplify the Expression

Carry out the multiplications: \( 2 \times 1 = 2 \) and \( 5 \times 1 = 5 \). Then add the results: \( 2 + 5 = 7 \).
05

Write the Simplified Result

The simplified expression is \( 7 \).

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Key Concepts

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

Zero Exponent Rule
The zero exponent rule is a fundamental concept in algebra that states any non-zero number raised to the power of zero equals one. This rule helps simplify expressions significantly. For example, in the equation given, we have terms like \( x^0 \) and \( y^0 \). Since both of these terms are raised to the power of zero, they can be simplified directly to 1. This is because:

\[ x^0 = 1 \]
\[ y^0 = 1 \]

Every time you encounter a term with an exponent of zero, remember you can replace that entire term with 1, making your expression easier to manage.
Substitution
Substitution is a powerful technique used to simplify expressions by replacing variables with known values. In the steps of our solution, substitution plays a key role right after we apply the zero exponent rule.

First, we identify the terms with exponents of zero. We then apply the zero exponent rule to get \( x^0 = 1 \) and \( y^0 = 1 \).

The next step is to substitute these values back into the original expression. This transforms the complex terms into simple ones.

So, our expression \( 2 \times x^0 + 5 \times y^0 \) becomes:

\[ 2 \times 1 + 5 \times 1 \]

This step makes further simplification straightforward, laying the groundwork for basic arithmetic operations.
Basic Arithmetic Operations
Once we have simplified and substituted the values in the expression, we can easily apply basic arithmetic operations to find the final result.

In our example:

First, multiply the numbers given in the simplified expression:

\[ 2 \times 1 = 2 \]
\[ 5 \times 1 = 5 \]

Then, add these products together:

\[ 2 + 5 = 7 \]

Understanding and correctly performing these basic arithmetic operations is crucial.
  • Start with multiplication from left to right.
  • Then follow it up with addition or subtraction as required.
This will ensure you simplify expressions correctly and efficiently. Ultimately, the complete simplified result in this example is \( 7 \).

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