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Chapter 4: Problem 51

Evaluate. $$ 10^{2}-5^{3} $$

Short Answer

Expert verified
The result of the expression is -25.

Step by step solution

01

Evaluate the Power of 10

Calculate the value of the expression \(10^2\). Recall that \(10^2\) means 10 multiplied by itself. \[10^2 = 10 \times 10 = 100\]
02

Evaluate the Power of 5

Calculate the value of the expression \(5^3\). Recall that \(5^3\) means 5 multiplied by itself three times. \[5^3 = 5 \times 5 \times 5 = 125\]
03

Subtract the Results

Subtract the result of \(5^3\) from the result of \(10^2\) to find the final answer. \[100 - 125 = -25\]

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Key Concepts

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

Power of a Number
When we talk about the "power of a number," we are referring to something called "exponents." Exponents tell us how many times to multiply a number by itself. For example, in the expression \(10^2\), the number 10 is the base, and the number 2 is the exponent. This means we multiply 10 by itself: \(10 \times 10 = 100\).

Exponents make it easy to handle large numbers. Instead of writing long multiplication, like \(10 \times 10\), we simply use \(10^2\).
  • A base raised to the power of 1 is the base itself. For example, \(10^1 = 10\).
  • A base raised to the power of 0 is always 1, regardless of the base value. For instance, \(10^0 = 1\).

Understanding the power of a number helps in simplifying calculations and expressing large or repeated multiplication effectively.
Subtraction in Arithmetic
Subtraction is a fundamental arithmetic operation. It involves taking one number away from another. In our problem, once we evaluate the powers, we perform subtraction. For example, subtracting 125 from 100 can be written as \(100 - 125\).

Subtraction works by removing the smaller number from the larger one to find the difference, but when the larger number is being subtracted, it leads to a negative number.
  • When subtracting 0 from any number, the result is the number itself. For example, \(10 - 0 = 10\).
  • When a number is subtracted from itself, the result is 0. For instance, \(10 - 10 = 0\).

Subtraction helps us find the difference between quantities and is key to solving equations and understanding relative sizes.
Negative Numbers
Negative numbers are numbers less than zero. They are written with a minus sign (-) in front. In our calculation, after subtracting \(5^3\) from \(10^2\), we end up with -25, which is a negative number.

Working with negative numbers is common in areas like temperatures, elevations, and even debts. Here's how negative numbers behave:
  • Adding a negative number is like subtraction. For example, adding -5 to 10 is \(10 + (-5) = 5\).
  • Subtracting a negative number is like addition. For example, subtracting -5 from 10 is \(10 - (-5) = 15\).
  • The product of two negative numbers is positive. For instance, \((-5) \times (-5) = 25\).

Negative numbers help us to express and work with values in contexts where decreases or reversals occur. They are vital for a complete understanding of arithmetic operations.

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Most popular questions from this chapter

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} 3 y-x=6 \\ 4 x+12 y=0 \end{array}\right. $$

Rewrite each sentence using mathematical symbols. Do not solve the equations. Twice a number, added to \(6,\) is 3 less than the number.

Suppose you are solving the system \(\left\\{\begin{array}{l}3 x+8 y=-5 \\ 2 x-4 y=3\end{array}\right.\) You decide to use the addition method by multiplying both sides of the second equation by 2 . In which of the following was the multiplication performed correctly? Explain. a. \(4 x-8 y=3\) b. \(4 x-8 y=6\)

Two occupations predicted to greatly increase in the number of jobs are pharmacy technicians and network system analysts. The number of pharmacy technician jobs predicted for 2006 through 2016 can be approximated by \(9.1 x-y=-295 .\) The number of network system analyst jobs predicted for 2006 through 2016 can be approximated by \(14 x-y=-262 .\) For both equations, \(x\) is the number of years since 2006 , and \(y\) is the number of jobs in the thousands. (Source: Bureau of Labor Statistics) a. Use the addition method to solve this system of equations. (Eliminate \(y\) first and solve for \(x\). Round this result to the nearest whole.) b. Interpret your solution from part (a). C. Using the year in your answer to part (b), estimate the number of pharmacy technician jobs and network system analyst jobs in that year.

For the years 1995 through \(2005,\) the annual percent \(y\) of U.S. households that used a wall or floor furnace to heat their houses is given by the equation \(y=-0.04 x+5.1,\) where \(x\) is the number of years after \(1995 .\) For the same period, the annual percent \(y\) of U.S. households that used fireplaces or wood stoves to heat their homes is given by \(y=-0.31 x+5.3\), where \(x\) is the number of years after 1995. (Source: U.S. Census Bureau, American Housing Survey Branch) a. Use the substitution method to solve this system of equations. $$ \left\\{\begin{array}{l} y=-0.04 x+5.1 \\ y=-0.31 x+5.3 \end{array}\right. $$ Round your answer to the nearest whole numbers. b. Explain the meaning of your answer to part (a). c. Sketch a graph of the system of equations. Write a sentence describing the use of wall furnaces or fireplaces or wood stoves for heating homes between 1995 and 2005 .
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