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Chapter 0: Problem 17

evaluate the expression for the given value of x. $$ 500 x^{60} \quad x=1.01 $$

Short Answer

Expert verified
To solve \(500x^{60}\) where \(x=1.01\), substitute \(1.01\) for \(x\) in the equation and then compute the resulting value.

Step by step solution

01

Identify the Expression and the Value of Variable

Here, the predetermined expression is \(500x^{60}\) and the value of x that needs to be substituted into the expression is \(1.01\).
02

Substituting the Given Variable Value

Replace every 'x' in the polynomial with \(1.01\). Now, the substituted expression should be \(500*(1.01)^{60}\).
03

Compute the Expression Value

Calculate the expression value by following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction). Here, first calculate the exponent \(1.01^{60}\), then multiply this result by \(500\).

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Key Concepts

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

Polynomial Evaluation
Polynomial evaluation is the process of determining the resulting value of a polynomial expression when specific values are substituted for its variables. This approach is essential in various mathematical and real-world applications, such as calculating numerical solutions in engineering and physics problems.To evaluate a polynomial:
  • First, substitute each variable in the polynomial with the specified values. In our example, we substituted \( x \) in the expression \( 500x^{60} \) with \( 1.01 \).
  • Second, perform the mathematical operations, often involving powers and arithmetic, to find the precise value of the expression. Proper use of mathematical tools or software can ensure accuracy, especially with large exponents, such as \( (1.01)^{60} \).
Polynomial evaluation helps deepen your understanding of how polynomial expressions behave when variables are given numerical values.
Order of Operations
Order of operations is a fundamental principle in mathematics that dictates the sequence in which calculations should be performed. It's especially important in evaluating expressions involving multiple operations like addition, subtraction, multiplication, and exponents.The order of operations is often remembered by the acronym PEMDAS:
  • Parentheses: Simplify expressions within parentheses first.
  • Exponents: Next, calculate any exponents.
  • Multiplication and Division: Perform these operations from left to right.
  • Addition and Subtraction: Finally, perform these operations from left to right.
In the given example, after substituting the value \( x = 1.01 \) into \( 500x^{60} \), you first need to compute \( 1.01^{60} \) due to the exponent. Only after calculating this power, should you proceed to multiply the result by 500. Following this order ensures that you arrive at the correct solution.
Exponents
Exponents represent repeated multiplication of a base number. Understanding how to handle exponents is crucial in evaluating polynomial expressions efficiently. In our example, the expression includes \( x^{60} \), which means multiplying \( x \) by itself 60 times.Key points about exponents:
  • An exponent specifies how many times the base number is used as a multiplier. For example, \( x^3 = x \times x \times x \).
  • Exponential growth can be very steep. As seen with \( 1.01^{60} \), even small base numbers result in significant increases when raised to large powers.
  • Scientific calculators or digital tools help compute large exponents accurately, as manual multiplication can be cumbersome.
Exponents allow compact expression of very large numbers or processes, making them integral to fields like algebra, finance, and science.

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Most popular questions from this chapter

evaluate the expression for the given value of x. $$ \frac{1+x^{-1}}{x^{-1}} \quad x=3 $$

find the domain of the given expression. $$ \sqrt{x^{2}+3} $$

Rationalize the numerator or denominator and simplify. $$ \frac{2}{\sqrt{10}} $$

simplify each expression by factoring. $$ \left(x^{4}+2\right)^{3}(x+3)^{-1 / 2}+4 x^{3}\left(x^{4}+2\right)^{2}(x+3)^{1 / 2} $$

a certificate of deposit has a principal of P and an annual percentage rate of r (expressed as a decimal) compounded n times per year. Enter the compound interest formula $$ A=P\left(1+\frac{r}{n}\right)^{N} $$ into a graphing utility and use it to find the balance after \(N\) compoundings. $$ P=\$ 7000, \quad r=5 \%, \quad n=365, \quad N=1000 $$
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