Problem 42 - Algebra 1: Concepts and Skills

Question

Simplify the variable expression. $$ \left(36 x^{3}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(6x^{3/2}\)

Step 1: Apply square root to each part of the expression separately

We have \((36x^{3})^{1/2}\), which can be rewritten as \(\sqrt{(36x^{3})} = \sqrt{36} \cdot \sqrt{x^{3}}\)

Step 2: Simplify the roots

After simplifying, we get \(6 \cdot x^{3/2}\). Here, we took the square root of \(36\) to get \(6\) and used that the square root of \(x^{n}\) is \(x^{n/2}\), hence \(x^{3/2}\)

Step 3: Final Answer

Therefore, the simplified version of the expression is \(6x^{3/2}\)

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It allows us to rewrite complex mathematical expressions in a simpler, more understandable form. The goal is to make solving problems more straightforward and efficient.
When simplifying expressions, we look for opportunities to combine like terms, factor out common elements, or apply mathematical properties. For instance, in the expression \((36x^{3})^{1/2}\), we can simplify it by using the properties of square roots and exponents.

• Identify Parts: Break down the expression into recognizable parts, like numbers and variables with exponents.
• Apply Properties: Use mathematical properties strategically, such as distributive, associative, and commutative laws.
• Combine and Simplify: Always check if there's more that can be combined or further simplified.

This process reduces the original expression to a cleaner, more workable form, as seen when \( (36x^{3})^{1/2} \) is simplified to \( 6x^{3/2} \).

Square Roots

The square root, denoted as \(\sqrt{\cdot}\), is a fundamental concept in mathematics. It represents a value that, when multiplied by itself, gives the original number.
For example, the square root of 36 is 6 because \(6 \times 6 = 36\). In algebra, square roots are frequently used to simplify expressions involving exponents.

• Basic Understanding: The square root of a number \(x\) is a number \(y\) such that \(y^2 = x\).
• Square Root of Variables: When dealing with variables and exponents, the square root operation divides the exponent by 2. Hence, \(\sqrt{x^n} = x^{n/2}\).

In our example, applying the square root to \(36x^3\) separately involved taking the square root of 36 and dividing the exponent of \(x\) by 2, resulting in \(6x^{3/2}\).

Exponents

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. They are crucial in simplifying and solving algebraic expressions.
When an expression has exponents, you can use the properties of exponents to simplify it. This includes multiplying exponents when bases are multiplied, adding when divisions occur, and raising a power to a power by multiplying the exponents.

• Multiplication Rule: \(x^a \times x^b = x^{a+b}\)
• Division Rule: \(x^a / x^b = x^{a-b}\)
• Power of a Power Rule: \((x^a)^b = x^{a \cdot b}\)
• Simplifying Square Roots with Exponents: When you take the square root of an exponent, you divide the exponent by 2.

In the step by step solution, dividing the exponent of \(x^3\) by 2 gives \(x^{3/2}\). This principle is a convenient tool for simplifying expressions, especially those involving square roots.