Problem 22
Question
Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(2 x+x^{2}\)
Step-by-Step Solution
Verified Answer
For (a) the answer is 24. For (b) the answer is 48.
1Step 1 - Substitute the value for part (a)
For part (a), substitute the value of \( x = 4 \) into the expression \( 2x + x^2 \).
2Step 2 - Simplify the expression for part (a)
After substitution for part (a):\(2(4) + 4^2\)Calculate the multiplication and exponentiation first:\(2 \cdot 4 = 8\) and \(4^2 = 16\)Then add the results: \(8 + 16 = 24\)
3Step 3 - Substitute the value for part (b)
For part (b), substitute the value of \( x = 6 \) into the expression \( 2x + x^2 \).
4Step 4 - Simplify the expression for part (b)
After substitution for part (b):\(2(6) + 6^2\)Calculate the multiplication and exponentiation first:\(2 \cdot 6 = 12\) and \(6^2 = 36\)Then add the results: \(12 + 36 = 48\)
Key Concepts
Substitution in AlgebraSimplifying ExpressionsExponentiation
Substitution in Algebra
Substitution in algebra is an essential concept, especially when evaluating expressions.
It involves replacing variables with their given numerical values.
In our example, we evaluated the expression \(2x + x^2\) for different values of \(x\): \(x = 4\) and \(x = 6.\) This means we substitute the given value of \(x\) into the expression wherever it appears.
Let's break it down:
1. For \(x = 4\), we replaced \(x\) with \(4\) in \(2x + x^2: 2(4) + 4^2\).
2. Similarly, for \(x = 6\), we replaced \(x\) with \(6\) in \(2x + x^2: 2(6) + 6^2\).
This critical step allows us to convert the algebraic expression into a numerical form, making calculations straightforward.
It involves replacing variables with their given numerical values.
In our example, we evaluated the expression \(2x + x^2\) for different values of \(x\): \(x = 4\) and \(x = 6.\) This means we substitute the given value of \(x\) into the expression wherever it appears.
Let's break it down:
1. For \(x = 4\), we replaced \(x\) with \(4\) in \(2x + x^2: 2(4) + 4^2\).
2. Similarly, for \(x = 6\), we replaced \(x\) with \(6\) in \(2x + x^2: 2(6) + 6^2\).
This critical step allows us to convert the algebraic expression into a numerical form, making calculations straightforward.
Simplifying Expressions
Simplifying expressions is a two-step process often needed after substitution.
The goal is to simplify the substituted expression into a single number.
In our example, we see this in action:
1. **Step 1: Perform multiplication**
For \(x = 4\), after substitution, the expression becomes \(2(4) + 4^2\).
Here, we perform the operation \(2 \times 4\), which equals \(8\).
2. **Step 2: Perform exponentiation**
Still for \(x = 4\), we calculate \(4^2\), which is \(16\).
Both these simplified results are then added together.
\(8 + 16 = 24\).
The process is the same for \(x = 6\). First, evaluate \(2(6)\) giving \(12\), and \(6^2\) giving \(36\). Adding these results: \(12 + 36 = 48\). By breaking it down into smaller steps, we make it easier to follow and understand.
The goal is to simplify the substituted expression into a single number.
In our example, we see this in action:
1. **Step 1: Perform multiplication**
For \(x = 4\), after substitution, the expression becomes \(2(4) + 4^2\).
Here, we perform the operation \(2 \times 4\), which equals \(8\).
2. **Step 2: Perform exponentiation**
Still for \(x = 4\), we calculate \(4^2\), which is \(16\).
Both these simplified results are then added together.
\(8 + 16 = 24\).
The process is the same for \(x = 6\). First, evaluate \(2(6)\) giving \(12\), and \(6^2\) giving \(36\). Adding these results: \(12 + 36 = 48\). By breaking it down into smaller steps, we make it easier to follow and understand.
Exponentiation
Exponentiation is a key part of algebra, representing how many times a number (the base) is multiplied by itself.
In our expression \(2x + x^2\), the \(x^2\) term uses exponentiation where the base is \(x\) and the exponent is \(2\).
Let's recap the exponentiation steps used in our example:
1. For \(x = 4\), we compute \(4^2\), which means multiplying \(4\) by itself: \(4 \times 4 = 16\).
2. For \(x = 6\), we compute \(6^2\), which means \(6 \times 6 = 36\).
Understanding this helps simplify the original expression accurately.
Always perform exponentiation before other operations like multiplication and addition unless parentheses dictate otherwise. This order of operations ensures correct results.
In our expression \(2x + x^2\), the \(x^2\) term uses exponentiation where the base is \(x\) and the exponent is \(2\).
Let's recap the exponentiation steps used in our example:
1. For \(x = 4\), we compute \(4^2\), which means multiplying \(4\) by itself: \(4 \times 4 = 16\).
2. For \(x = 6\), we compute \(6^2\), which means \(6 \times 6 = 36\).
Understanding this helps simplify the original expression accurately.
Always perform exponentiation before other operations like multiplication and addition unless parentheses dictate otherwise. This order of operations ensures correct results.
Other exercises in this chapter
Problem 22
In each term, give the numerical coefficient. \(9 n^{6}\)
View solution Problem 22
Determine whether each statement is true or false. Every terminating decimal is a rational number.
View solution Problem 22
Find each product. \(-4.6(0.24)\)
View solution Problem 23
Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ 4+15=15+4 $$
View solution