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Chapter 1: Problem 5

Perform each operation. $$\frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3}$$

Short Answer

Expert verified
The result of \(\frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3}\) is \(\frac{8}{27}\).

Step by step solution

01

Identify the Numerator and the Denominator

The numerator, which is the top part of the fraction, is each case '2'. The denominator, the bottom part of the fraction, is '3'.
02

Multiply the Numerators

Multiply the three numerators together: \(2 \cdot 2 \cdot 2 = 8\). So, the numerator of the result will be '8'.
03

Multiply the Denominators

Multiply the three denominators together: \(3 \cdot 3 \cdot 3 = 27\). So, the denominator of the result will be '27'.

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Key Concepts

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

Fractions
In the world of mathematics, a fraction represents a part of a whole. It verges on the magic of division, allowing us to express quantities that are not whole numbers. It's the slice of pizza you indulge in, or the half of an apple you share.

Envision a fraction as a delightful pie sliced into equal parts. The pieces signify the fractions鈥攚ith the top number, called the numerator, telling you how many pieces you've got, and the bottom number, called the denominator, conveying how many pieces the pie was cut into in the first place. Multiply fractions, and you're essentially combining these slices in a new way to see what portion of the pie you end up with.
Numerator and Denominator
Ah, the numerator and the denominator, the dynamic duo of the fraction world! They are the 'yin and yang' that balance each fraction. The numerator is the number above the fraction line, showing exactly how many parts we are focused on. Imagine you're holding balloons鈥攖his number tells how many are in your hands.

The denominator, on the other hand, is the anchor beneath the fraction line, holding down the fort. It decides the size of each part by telling us into how many equal slices the whole is divided. If the whole cake is cut into 8 pieces, then 8 is your denominator, guiding you on the size of each mouthful.
Arithmetic Operations with Fractions
When it comes to arithmetic operations with fractions, think of it as an intriguing puzzle. Multiplying fractions is just one piece of this puzzle, and it's simpler than you might imagine. To multiply fractions, you just multiply the numerators together to get a new numerator, and then do the same with the denominators to get a new denominator.

Let's not forget that sometimes, you can simplify the multiplied fraction to make it easier on the eyes and brain! Simplifying, or reducing fractions, means to transform them to the most streamlined version possible鈥攋ust like decluttering your room to enjoy more free space.

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Most popular questions from this chapter

Let \(x=2, y=-3,\) and \(z=1 .\) Show that the two expressions have the same value. $$(x+y)+z ; y+(x+z)$$

Write each inequality as an equivalent inequality in which the inequality symbol points in the opposite direction. $$3 \leq 7$$

Which property of real numbers justifies each statement? $$3+x=x+3$$

Write each inequality as an equivalent inequality in which the inequality symbol points in the opposite direction. $$8 \cdot 2 \geq 8 \cdot 1$$

Explain why division is not commutative.
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