Problem 40 - Intermediate Algebra

Question

Simplify. \(\sqrt{64 y^{9}}\)

Step-by-Step Solution

Answer

The simplified expression is \(8y^4\sqrt{y}\).

Step 1: Understand the Expression

We are given the expression \(\sqrt{64 y^9}\). Our goal is to simplify it by removing the square root as much as possible.

Step 2: Split the Square Root

The square root of a product can be split into the product of square roots, i.e., \(\sqrt{64 y^9} = \sqrt{64} \cdot \sqrt{y^9}\). This allows us to handle the numerical and variable parts separately.

Step 3: Simplify the Numerical Part

The square root of 64 is a number that, when multiplied by itself, equals 64. Since \(8 \times 8 = 64\), \(\sqrt{64} = 8\).

Step 4: Simplify the Variable Part

Next, simplify \(\sqrt{y^9}\). Remember that \(\sqrt{y^9} = (y^9)^{1/2}\). To simplify, we use the rule: \((x^m)^n = x^{m \cdot n}\). Thus, \((y^9)^{1/2} = y^{9/2}\).

Step 5: Express in Radical Form

Convert \(y^{9/2}\) into radical form. The easiest way is to express it as \(y^{4.5} = y^4 \times y^{0.5} = y^4 \times \sqrt{y}\).

Step 6: Combine Simplified Parts

Now that we have \(8\) from \(\sqrt{64}\) and \(y^4 \times \sqrt{y}\) from \(\sqrt{y^9}\), we can write the simplified expression as \(8y^4\sqrt{y}\).

Key Concepts

Understanding Square Roots

Square roots are an essential concept in mathematics and are particularly important when simplifying expressions. The square root of a number is another number that, when squared, equals the original number. For example, the square root of 64 is 8 because

• 8 multiplied by 8 equals 64.

This operation is represented using the radical symbol, like so:

• \( \sqrt{64} = 8 \).

Square roots can be applied to both numbers and algebraic terms, making them versatile in solving various types of problems. When simplifying expressions, it is crucial to split the square root if the expression is a combination of numerical constants and variables, as demonstrated in the exercise

• \( \sqrt{64 y^9} = \sqrt{64} \cdot \sqrt{y^9} \).

Exploring Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. For example,

• \( y^9 \) indicates that the base \( y \) is multiplied by itself 9 times.

Exponents are a powerful tool when simplifying expressions, especially within square roots. They allow us to transform expressions for easier manipulation. For instance, when we have a square root involving an exponent like

• \( \sqrt{y^9} = (y^9)^{1/2} \),

we apply the rule of exponents

• \( (x^m)^n = x^{m \cdot n} \)

to further simplify. Thus,

• \( (y^9)^{1/2} = y^{9/2} \).

Understanding this principle aids in breaking down complex expressions into parts that are easier to work with.

Understanding Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. The radical symbol \( \sqrt{} \) is commonly used to denote a square root. Simplifying radical expressions involves expressing them in a form that is more compact or that removes the radical where possible.

• For example, \( \sqrt{y^9} \) can be rewritten as a radical expression because \( y^9 \) raised to the 1/2 power simplifies to \( y^{9/2} \).

In practice, you may convert expressions between radical and exponent notations to ease calculations. In our example, \( y^{9/2} \) can be expressed as \( y^4 \cdot \sqrt{y} \).

• This approach demonstrates how exponents can simplify radicals.

The goal is to make the expression more straightforward while still equivalent to the initial form.

Strategies for Algebraic Simplification

Algebraic simplification is the process of transforming expressions into their simplest or most efficient form. This involves applying rules for exponents, breaking down radicals, and manipulating terms to achieve a more straightforward expression. In the original exercise, our primary goal was to simplify

• \( \sqrt{64 y^9} \).

By dealing with numerical parts first, we acknowledged that

• \( \sqrt{64} = 8 \),

then focused on simplifying the variable part. After expressing powers in a simplified form, such as

• \( y^{9/2} = y^4 \times \sqrt{y} \),

we combined all simplified components together. The final expression,

• \( 8y^4 \sqrt{y} \),

almost always represents a neater and more manageable form, serving as our simplified solution. Mastery of these strategies enables you to simplify increasingly complex expressions by breaking them down into their component parts and reducing potential errors.