Problem 33 - Algebra 2

Question

Simplify. \(\sqrt{\frac{4 r^{8}}{t^{9}}}\)

Step-by-Step Solution

Answer

\( \frac{2r^4}{t^{9/2}} \)

Step 1: Simplify the Square Root

The expression to simplify is \( \sqrt{\frac{4 r^{8}}{t^{9}}} \). First, recall the property of square roots which states \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Applying this property:\[\sqrt{\frac{4 r^{8}}{t^{9}}} = \frac{\sqrt{4r^8}}{\sqrt{t^9}}\]

Step 2: Simplify the Numerator's Square Root

Now, simplify \( \sqrt{4r^8} \). The square root of a product is the product of the square roots: \( \sqrt{4} \times \sqrt{r^8} \). \( \sqrt{4} = 2 \) since 2 times 2 is 4, and \( \sqrt{r^8} = r^{8/2} = r^4 \).Thus, \( \sqrt{4r^8} = 2r^4 \).

Step 3: Simplify the Denominator's Square Root

Next, simplify \( \sqrt{t^9} \). Using the property of square roots for exponents, \( \sqrt{t^9} = t^{9/2} \).

Step 4: Write the Simplified Expression

Combine the results from step 2 and step 3 to write the simplified expression. \[\frac{\sqrt{4r^8}}{\sqrt{t^9}} = \frac{2r^4}{t^{9/2}}\]

Step 5: Final Simplified Expression

The expression is now simplified and can be written as: \[\frac{2r^4}{t^{4.5}}\]Or, equivalently, in fractional exponent form:\[\frac{2r^4}{t^{9/2}}\]

Key Concepts

Understanding the Property of Square Roots

Square roots have unique properties that allow us to simplify expressions effectively. One crucial property is

• If you have a square root of a fraction, it can be separated into the square root of the numerator and the square root of the denominator. For instance, if you see \( \sqrt{\frac{a}{b}} \), it can be rewritten as \( \frac{\sqrt{a}}{\sqrt{b}} \). This property helps break down complex expressions for easier computation.

Applied to our example,\( \sqrt{\frac{4 r^{8}}{t^{9}}} \) can be simplified to \( \frac{\sqrt{4r^8}}{\sqrt{t^9}} \). By doing this, you can handle the numerator and the denominator separately, streamlining the simplification process.

Applying Exponent Rules

Exponent rules are essential for managing expressions with variables raised to powers. These rules help us manipulate and simplify terms effectively:

• Product of Powers: When multiplying like bases, add their exponents. For example, \( x^a \times x^b = x^{a+b} \).
• Power of a Power: When raising an exponent to another power, multiply the exponents. For instance, \( (x^a)^b = x^{a \cdot b} \).
• Power of a Product: Any product raised to an exponent means each factor is individually raised to that exponent: \( (ab)^c = a^c \times b^c \).
• Root as a Fractional Exponent: Any nth root of a number equals raising that number to the fraction \( \frac{1}{n} \).

For example, in \( \sqrt{r^8} \), it can be seen as \( (r^8)^{1/2} \), simplifying to \( r^{8/2} = r^4 \). Exponent rules make such manipulation clear and systematic.

Mastering Simplifying Radicals

Simplifying radicals involves breaking down a square root or any other root into its simplest form. The key is to look for perfect squares or other perfect powers within the radicand (the number inside the root) and simplify step-by-step:

• Break down the number or expression under the root into factors.
• Find any perfect squares, cubes, or higher powers present, since these can be taken out of the root.
• For numbers, identify the largest perfect square factor. For variables, use exponent rules to simplify.

In the example \( \sqrt{4r^8} \), recognizing \( 4 \) as \( 2^2 \) gives \( \sqrt{4} = 2 \). Similarly, identify \( r^8 \) as a perfect square \((r^4)^2\), reducing the root to \( r^4 \). This skill simplifies expressions for easier calculation and understanding.

Exploring Fractional Exponents

Fractional exponents transform roots into a form more convenient for algebraic manipulation. Here's how they work:

• Definition: The nth root of a number, \( x^{1/n} \), is equivalent to \( \sqrt[n]{x} \). For example, \( x^{1/2} = \sqrt{x} \).
• Combining Rules: Multiply roots using their fractional exponents. If \( x^{a/n} \times x^{b/n} \), then it equals \( x^{(a+b)/n} \).
• Simplifying Expressions: Use fractional exponents to simplify exponential and radical expressions. They make it easier to apply standard arithmetic operations.

For example, \( t^{9/2} \) in the expression accounts for \( \sqrt{t^9} \), leading to a more straightforward final expression \( \frac{2r^4}{t^{9/2}} \), equivalent to \( \frac{2r^4}{t^{4.5}} \). Understanding fractional exponents gives a strong toolset for dealing with more complex algebraic expressions efficiently.