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Chapter 5: Problem 104

How does doubling a number affect its square root?

Short Answer

Expert verified
Doubling a number and then finding its square root gives a result that is \( \sqrt{2} \) times the square root of the original number.

Step by step solution

01

Recognize the Problem

We are tasked to find the effect of doubling a number on its square root. Let's represent our original number as \( x \) and the doubled number as \( 2x \). We aim to find the relationship between \( \sqrt{x} \) (the square root of the original number) and \( \sqrt{2x} \) (the square root of the doubled number).
02

Calculate the Square Root of the Doubled Number

First, note that doubling a number and taking the square root are two separate, independent operations. So, to calculate the square root of \( 2x \), we break down \( 2x \) under the square root sign to \( \sqrt{2} * \sqrt{x} \). We do this because the square root of a product equals the product of the square roots.
03

Compare the Results

Now that we know \( \sqrt{2x} = \sqrt{2} * \sqrt{x} \), we can clearly see that doubling a number and then finding its square root gives a value that is \( \sqrt{2} \) times larger than the square root of the original number.

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Key Concepts

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

Doubling Numbers
Doubling a number simply means multiplying it by 2. Imagine starting with a number like 5. When you double it, you end up with 10. This is a straightforward operation that involves simple multiplication.

In algebra, if we start with a variable representation like \( x \), doubling that number will give us \( 2x \).

Here's a quick recap of steps to double a number:
  • Identify the original number.
  • Multiply it by 2.
  • That's your doubled number!
Doubling affects calculations such as finding the square root, where properties change as numbers are manipulated.
Understanding how doubling interacts with other operations like square roots can deepen your mathematical insight.
Algebraic Expressions
Algebraic expressions are a way of representing numbers or relations in terms of variables and constants. They typically involve operations like addition, subtraction, multiplication, and division.

If we take our expression of doubling a number, \( x \), in algebraic terms, it becomes \( 2x \). This expression is a tidy representation showing us 'double' of whatever \( x \) stands for.

Let鈥檚 break down an algebraic expression:
  • Variables: These are symbols (like \( x \) or \( y \)) that represent unknown numbers.
  • Constants: Fixed numbers that do not change, such as 2 or 3.
  • Operations: Includes addition (+), subtraction (-), multiplication (\( \times \)), and division (\( \div \)).

Working with algebraic expressions involves simplifying or solving by following the rules of operations while respecting the order in which they are executed.
This simplifies complex problems into manageable steps.
Mathematical Operations
Mathematical operations are the actions taken to solve a problem. When dealing with expressions like doubling a number and taking square roots, it鈥檚 crucial to understand each operation's role.

Let's first look at multiplication which, when doubled, is simply multiplying the original number by 2. Next is square roots, a bit more complex, involving reversing the process of squaring. The square root of a number \( x \), written as \( \sqrt{x} \), is that number which, when multiplied by itself, gives \( x \).
This can be seen in common operations such as:
  • Finding the square root of \( 2x \) involves breaking it into \( \sqrt{2} \times \sqrt{x} \).
  • Understanding that \( \sqrt{2} \) is a constant that approximately equals 1.414, hence scaling the result.
In our doubling context when we go from \( x \) to \( 2x \), the square root sequence changes proportionally, showing a direct multiplicative effect. Each mathematical operation aligns with particular rules and properties, vital for arriving at correct solutions.

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Most popular questions from this chapter

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of Texas for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 20.85 & 21.27 & 21.70 & 22.13 & 22.57 & 23.02 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 23.48 & 23.95 & 24.43 & 24.92 & 25.15 \\ \hline \end{array} \end{aligned} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling Texas's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project Texas's population, in millions, for the year 2020 . Round to two decimal places.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Subtraction is a commutative operation.

Use a calculator with a square root key to find a decimal approximation for each square root. Round the number displayed to the nearest \(\mathbf{a}\). tenth, b. hundredth, c. thousandth. \(\sqrt{3176}\)

Shown in the figure is a 7-hour clock and the table for clock addition in the 7-hour clock system. $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \oplus & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline \mathbf{0} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \mathbf{1} & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\ \hline \mathbf{2} & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\ \hline \mathbf{3} & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\ \hline \mathbf{4} & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\ \hline \mathbf{5} & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\ \hline \mathbf{6} & 6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{array} $$ a. How can you tell that the set \(\\{0,1,2,3,4,5,6\\}\) is closed under the operation of clock addition? b. Verify one case of the associative property: \((3 \oplus 5) \oplus 6=3 \oplus(5 \oplus 6)\) c. What is the identity element in the 7-hour clock system? d. Find the inverse of each element in the 7-hour clock system. e. Verify two cases of the commutative property: \(4 \oplus 5=5 \oplus 4\) and \(6 \oplus 1=1 \oplus 6\)

Reduce each rational number to its lowest terms. \(\frac{60}{108}\)
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