Problem 110 - Elementary and Intermediate Algebra

Question

Look Alikes \(\cdots\) a. \(\sqrt[3]{27}-5 \sqrt[3]{8}\) b. \(\sqrt{27}-5 \sqrt{8}\)

Step-by-Step Solution

Answer

a: -7 b: \(3\sqrt{3} - 10\sqrt{2}\)

Step 1: Simplify Cube Root Expressions

First, simplify the cube roots in expression (a). The cube root of 27, written as \( \sqrt[3]{27} \), is 3 because \( 3^3 = 27 \). Similarly, the cube root of 8, \( \sqrt[3]{8} \), is 2 because \( 2^3 = 8 \). Thus, the expression becomes \( 3 - 5 \times 2 \).

Step 2: Calculate Expression (a)

Now substitute the simplified values into expression (a) from Step 1. Calculate \( 3 - 5 \times 2 \). Multiply 5 by 2 to get 10, and then subtract: \( 3 - 10 = -7 \).

Step 3: Simplify Square Root Expressions

Next, focus on simplifying the square roots in expression (b). The square root of 27, written as \( \sqrt{27} \), can be broken into \( \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3 \sqrt{3} \). Similarly, the square root of 8, \( \sqrt{8} \), can be broken into \( \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2} \). So, expression (b) becomes \( 3\sqrt{3} - 5 \times 2\sqrt{2} = 3\sqrt{3} - 10\sqrt{2} \).

Step 4: Combine and Retain Expression (b)

Simplified expression (b), \( 3\sqrt{3} - 10\sqrt{2} \), cannot be further simplified because \( \sqrt{3} \) and \( \sqrt{2} \) are irrational numbers, and you cannot combine them to produce a simpler form. So, the final answer for (b) remains as \( 3\sqrt{3} - 10\sqrt{2} \).

Key Concepts

Cube Roots

Understanding cube roots is important when simplifying expressions with radical terms. A cube root of a number is a value that, when multiplied by itself three times, gives the original number back. For example, the cube root of 27—written as \( \sqrt[3]{27} \)—is 3. That's because multiplying 3 by itself twice more (\( 3 \times 3 \times 3 \)) equals 27.

Here are some basics to keep in mind:

• Cubic numbers are numbers like 1, 8, 27, and 64. The cube root of these numbers will be integers. These numbers are called perfect cubes.
• If the cube root isn't a whole number, you might be dealing with an approximation or decimal.
• When simplifying expressions that contain cube roots, factor the number into its smallest cube factors, if possible, to easily identify the cube root.

Having a strong grasp of cube roots helps make solving radical expressions much simpler.

Square Roots

Square roots can seem similar to cube roots, but they involve a number multiplied by itself once to result in the original number. For instance, the square root of 9, denoted as \( \sqrt{9} \), is 3 because \( 3 \times 3 = 9 \).

A few key points about square roots:

• Perfect square numbers include 1, 4, 9, 16, 25, and so on. These have whole numbers as their square roots.
• If a number is not a perfect square, its square root will be an irrational number—this means it cannot be expressed as a simple fraction.
• A common method to simplify non-perfect square roots is to break them down into a product of perfect squares and other numbers. This reduces the square root into simpler terms. For example, \( \sqrt{27} \) can be seen as \( \sqrt{9 \times 3} \), which simplifies to \( 3 \sqrt{3} \).

Simplifying square roots is about recognizing and using perfect squares wherever possible.

Simplification Process

Simplifying expressions with roots, whether square or cube, involves breaking them down to their most basic parts. This process can seem confusing, but it becomes straightforward with practice and patience.

Here’s how to work through simplifications:

• Identify Perfect Powers: Always look for perfect squares or cubes within the number under the root, as this makes simplification easier.
• Factorization: Break the number into its prime factors when dealing with non-perfect roots, and find any perfect powers you can take out of the root first.
• Combine Like Terms: After simplifying the roots, combine any like terms. This may involve operations such as multiplication or subtraction.
• Check Your Work: Ensure that results are fully simplified and verify through reverse calculations if necessary.

Simplification is about reducing expressions to their simplest form without altering their values. Practice makes perfect in mastering this fundamental math skill.