Problem 18 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{8 x^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \(2x \sqrt{2x}\).

Step 1: Break down the expression inside the square root

We begin by factoring the expression inside the square root. The expression given is \(8x^3\). We can break down the number 8 as \(2^3\), and we express \(x^3\) as \((x^2) \cdot x\). So now we have \(8x^3 = 2^3 \cdot x^2 \cdot x\).

Step 2: Apply the square root to each factor

Next, we apply the square root to each factor separately: \(\sqrt{2^3} \cdot \sqrt{x^2} \cdot \sqrt{x}\).

Step 3: Simplify each square root

- The square root of \(2^3\) is \(\sqrt{2^3} = 2 \sqrt{2}\) because \(\sqrt{2^2} = 2\) and \(\sqrt{2}\) remains under the radical.- The square root of \(x^2\) is \(x\) because \(\sqrt{x^2} = x\).- The square root of \(x\) remains \(\sqrt{x}\), as it cannot be simplified further.

Step 4: Combine the simplified parts

Now, combine all the simplified parts: From Step 3, we get \(2 \sqrt{2} \cdot x \sqrt{x}\). Combine the constants and variables outside the radicals, and multiply the expressions inside the radicals: \(2x \sqrt{2x}\).

Key Concepts

Square Roots

The concept of square roots is fundamental in mathematics and involves finding a number which, when multiplied by itself, gives the original number. The square root of a number \( n \) is usually represented as \( \sqrt{n} \). This is especially useful when simplifying expressions under square roots or radicals. When given a complex expression like \( \sqrt{8x^3} \), breaking it into simpler parts helps achieve simplification. For instance, in the example expression, 8 is factored as \( 2^3 \), allowing easy application of the square root laws to each component. This division into separate factors simplifies the understanding and manipulation of square roots.

Factoring

Factoring is the process of breaking down numbers or expressions into multiple factors that, when multiplied together, will give the original number or expression. This is crucial when dealing with radicals as it simplifies complex expressions by identifying common components.

• In our example, \( 8x^3 \) was factored into \( 2^3 \cdot x^2 \cdot x \).
• Factoring makes it easier to apply operations like taking the square root to each component separately.

By learning to factor expressions efficiently, one can simplify and solve more advanced problems effectively, making the overall solution approach more manageable.

Expression Simplification

Expression simplification is the process of reducing a mathematical expression to its simplest form. This involves applying arithmetic operations and laws of mathematics such as factoring and dealing with exponents.

• First, break down complex expressions into simpler parts, as done in the step for \( 8x^3 \), simplifying it to \( 2 \sqrt{2} \cdot x\sqrt{x} \).
• Identify and group like terms or simplifiable components.

Simplifying an expression not only makes it easier to work with but also helps in understanding the underlying mathematical relationships and operations, aiding further calculations with accuracy.

Exponents and Powers

Exponents represent the number of times a number, known as the base, is multiplied by itself. They are expressed in the form \( a^n \), where \( a \) is the base and \( n \) is the exponent.

• In simplification, recognizing powers, such as \( x^3 \) can be split into \( x^2 \cdot x \), is key.
• Understanding that \( \sqrt{x^2} = x \) connects exponents with radicals and is used to simplify the expression \( x^3 \) under a square root.

Effective manipulation of expressions with exponents is enhanced by recognizing patterns and using the laws of exponents, thereby facilitating simpler calculations and achieving solutions with efficiency.