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Chapter 5: Problem 16

Evaluate each expression with the given replacement values. $$ 4 x^{2} \text { when } x=5 $$

Short Answer

Expert verified
The expression evaluates to 100.

Step by step solution

01

Substitute the Value of x

The problem provides the expression \( 4x^2 \) and asks you to evaluate it when \( x = 5 \). Start by substituting the value of \( x \) into the expression. So, we have: \( 4(5)^2 \).
02

Evaluate the Power

Next, evaluate the exponent. Here, \( (5)^2 \) means \( 5 \times 5 \), which equals \( 25 \). Now, the expression becomes \( 4 \times 25 \).
03

Perform Multiplication

Finally, multiply \( 4 \) by \( 25 \) to find the value of the expression: \( 4 \times 25 = 100 \).
04

Conclusion

Having performed all the calculations, we can conclude that the value of the expression \( 4x^2 \) when \( x=5 \) is \( 100 \).

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Key Concepts

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

Substitution in Algebra
Substitution in algebra is simply replacing a variable with a given value. This technique allows you to evaluate expressions when specific variable values are provided. Imagine each letter in the algebraic expression acts like a box, and you're told what number to put in the box. For instance, in our example, the expression is given as \( 4x^2 \) and \( x \) is replaced by \( 5 \).

When substituting:
  • Identify the variable you need to replace (here, it's \( x \)).
  • Replace every occurrence of this variable with the given number (here, substitute \( 5 \) everywhere you see \( x \)).
This results in the expression \( 4(5)^2 \). Substitution helps make abstract algebraic expressions concrete, allowing you to perform actual calculations.
Exponents in Algebra
Exponents in algebra represent repeated multiplication. If you see an expression like \( x^2 \), it means you need to multiply \( x \) by itself. So \( x^2 = x \times x \). Exponents are a way to simplify expressions, especially when numbers are multiplied several times.

In the example, we have \( (5)^2 \). This simplification means:
  • Multiply \( 5 \) by itself: \( 5 \times 5 \).
  • This results in \( 25 \).
Understanding exponents can simplify complex calculations, making patterns more visible and problems easier to solve. They're crucial in algebra for managing larger powers and making arithmetic manageable.
Multiplication in Algebra
Multiplication in algebra involves handling numbers or expressions through repeated addition or using shorthand calculations. It plays a significant role in combined operations within expressions.

In the given expression \( 4 \times 25 \), we choose to multiply after evaluating any exponents. Follow these steps:
  • The number \( 4 \) is multiplied by the result of the exponentiation (\( 25 \)).
  • This leads to: \( 4 \times 25 = 100 \).
Algebra uses multiplication to succinctly express repeated addition or combine terms in a straightforward manner. Mastery of multiplication helps in unraveling and simplifying complex equations efficiently. Remember that in algebra, multiplication is often indicated without a multiplication sign, especially when variables and coefficients are involved.

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Most popular questions from this chapter

Subtract. $$ (5 x+8)-\left(-2 x^{2}-6 x+8\right) $$

Subtract \(\left(3 x^{2}-4\right)\) from the sum of \(\left(x^{2}-9 x+2\right)\) and \(\left(2 x^{2}-6 x+1\right)\)

The polynomial \(-0.92 x^{2}+2.43 x+34.85\) represents the number of Americans (in millions) under age 65 covered by public health programs during \(1999-2007\). The polynomial \(0.07 x^{2}-0.64 x+180.96\) represents the number of Americans (in millions) under age 65 covered by private health insurance during 1999-2007. In both polynomials, \(x\) represents the number of years since \(1999 .\) Find a polynomial for the total number of Americans (in millions) under age 65 with some form of health coverage during this period. (Source: Based on data from U.S. Census Bureau)

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