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Chapter 11: Problem 10

Evaluate each of the numerical expressions. $$\frac{\sqrt{144}}{\sqrt[3]{8}}$$

Short Answer

Expert verified
The value of the expression is 6.

Step by step solution

01

Understanding the Problem

The problem requires us to evaluate the expression \( \frac{\sqrt{144}}{\sqrt[3]{8}} \). This means we need to calculate the square root of 144 and the cube root of 8 and then divide the results.
02

Calculate the Square Root

The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). Here, we need \( \sqrt{144} \). Calculating gives \( \sqrt{144} = 12 \), because \( 12 \times 12 = 144 \).
03

Calculate the Cube Root

The cube root of a number \(x\) is a value that, when used three times in a multiplication, gives \(x\). Here, we need \( \sqrt[3]{8} \). Calculating gives \( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \).
04

Perform the Division

Now that we have \( \sqrt{144} = 12 \) and \( \sqrt[3]{8} = 2 \), we substitute these values into the expression: \( \frac{\sqrt{144}}{\sqrt[3]{8}} = \frac{12}{2} \). Simplifying this fraction gives the result: \( 6 \), since \( 12 \div 2 = 6 \).

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Key Concepts

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

Square Roots
The square root is a concept in mathematics that involves finding a number which, when multiplied by itself, results in a given number. If you have a number \( x \), its square root is denoted as \( \sqrt{x} \). For example, the square root of 144 is 12 because \( 12 \times 12 = 144 \). This means that 12 is the number which gives us 144 when squared.
  • Square roots are used in various fields such as algebra, geometry, and calculus.
  • It's important to note that every positive number has two square roots: one positive and one negative. For example, -12 is also a square root of 144.
Another key aspect of square roots is that they are closely related to exponents. In mathematical terms, the square root of a number \( x \) can be written as \( x^{\frac{1}{2}} \). This notation emphasizes the idea of reversing the action of squaring a number.
Cube Roots
Cube roots extend the idea of finding roots to the power of three. The cube root of a number \( x \), denoted as \( \sqrt[3]{x} \), is a number \( y \) such that \( y \times y \times y = x \). For instance, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).
  • Cube roots can be found for both positive and negative numbers. For a negative number, the cube root is also negative. For example, the cube root of -8 is -2.
  • Cube roots are useful in solving equations where variables are raised to the third power and in geometric problems involving volume.
In terms of exponents, cube root is equivalent to raising a number to the power of \( \frac{1}{3} \). This alternative expression can be helpful in algebraic manipulations and calculations.
Division
Division is a fundamental arithmetic operation that involves splitting a number into equal parts. In expressions, division is often represented by the symbol \( \div \) or as a fraction. For example, \( 12 \div 2 \) is the same as \( \frac{12}{2} \). The goal of division is to determine how many times one number (the divisor) fits into another (the dividend).
  • In the context of the original exercise, division is the final step, where we divide the result of the square root of 144 by the cube root of 8.
  • When dividing by fractions, it's useful to remember to "multiply by the reciprocal." This means flipping the fraction of the divisor and then multiplying.
  • Division by zero is undefined, meaning you can't divide any number by zero. This is a crucial rule to remember in mathematics.
In equations, division can also be reversed using multiplication. This relationship helps in solving algebraic equations quickly and is a basic yet powerful concept in math understanding.

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Most popular questions from this chapter

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ -5 i(7-8 i) $$

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ (-5-8 i)(-5+8 i) $$

For Problems \(13-34\), add or subtract the complex numbers as indicated. $$ (-1-2 i)+(-6-6 i) $$

The function \(f(\mathrm{C})=\frac{9}{5} \mathrm{C}+32\) expresses the temperature in degrees Fahrenheit as a function of the temperature in degrees Celsius. (a) Use the function to complete the table. $$ \begin{array}{l|l|l|l|l|l|l|l} \hline \mathbf{C} & 0 & 10 & 15 & -5 & -10 & -15 & -25 \\ \hline \boldsymbol{f}(\mathrm{C}) & & & & & & & \\ \hline \end{array} $$ (b) Graph the linear function \(f(\mathrm{C})=\frac{9}{5} \mathrm{C}+32\). (c) Use the graph from part (b) to approximate the temperature in degrees Fahrenheit when the temperature is \(20^{\circ} \mathrm{C}\). Then use the function to find the exact value.

Solve each of the following quadratic equations, and check your solutions. $$n^{2}-4 n+5=0$$
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