Problem 78 - Algebra A Combined Function

Question

Assume that all variables represent positive real numbers. $$ \sqrt{\frac{y^{10}}{9 x^{6}}} $$

Step-by-Step Solution

Answer

\( \frac{y^{5}}{3x^{3}} \)

Step 1: Understanding the Square Root

The expression inside the square root is \( \frac{y^{10}}{9x^{6}} \). This means you need to find the square root of both the numerator \( y^{10} \) and the denominator \( 9x^{6} \).

Step 2: Simplifying the Numerator

Take the square root of the numerator \( y^{10} \). As per the properties of exponents, the square root is equivalent to raising the exponent to \( \frac{1}{2} \). So, \( \sqrt{y^{10}} = y^{10 \times \frac{1}{2}} = y^{5} \).

Step 3: Simplifying the Denominator

Take the square root of the denominator \( 9x^{6} \). The square root of 9 is 3, and for \( x^{6} \), it is \( x^{6 \times \frac{1}{2}} = x^{3} \). Therefore, \( \sqrt{9x^{6}} = 3x^{3} \).

Step 4: Combining Results

Now combine the results from the numerator and the denominator: \( \frac{y^{5}}{3x^{3}} \). Thus, the expression becomes \( \frac{y^{5}}{3x^{3}} \).

Key Concepts

Square Root

The square root is a concept that refers to finding a number which, when multiplied by itself, gives the original number. In mathematical terms, for any non-negative number \( a \), the square root of \( a \) is denoted as \( \sqrt{a} \). The square root operation is pivotal in simplifying expressions, especially when dealing with exponential terms.
When you see a square root symbol like \( \sqrt{\frac{y^{10}}{9x^{6}}} \), your goal is to simplify both the numerator \( y^{10} \) and the denominator \( 9x^{6} \) separately to make the expression less complex.

• For any expression \( \sqrt{a^{b}} \), it can also be expressed with exponents as \( a^{b/2} \).
• If \( a = b \times b \), then the square root of \( a \) is \( b \).

Breaking down expressions in this way is fundamental in simplifying complex algebraic structures.

Properties of Exponents

Exponents are shorthand for repeated multiplication, and they have several key properties that help simplify expressions, especially when they are part of square roots. Knowing these properties can make it much easier to handle algebraic problems:

• Product of Powers: \( a^{m} \times a^{n} = a^{m+n} \) - When multiplying with the same base, add the exponents.
• Power to a Power: \( (a^{m})^{n} = a^{m \times n} \) - Multiply the exponents when raising a power to another power.
• Quotient of Powers: \( \frac{a^{m}}{a^{n}} = a^{m-n} \) - When dividing with the same base, subtract the exponents.

In the context of square roots, the property \( a^{m/n} \) denotes the \( n \)-th root of \( a \) raised to the \( m \) power, which is essential for simplifying expressions like \( y^{10} \) and \( x^{6} \) encountered in the exercise.

Simplifying Expressions

Simplifying expressions is the process of reducing an algebraic expression to its simplest form. This typically involves removing any unnecessary components, such as constants or like terms, to make the expression as straightforward as possible.
In our exercise, the expression \( \sqrt{\frac{y^{10}}{9x^{6}}} \) was simplified to \( \frac{y^{5}}{3x^{3}} \) by:

• Applying the square root to each part separately—both to the numerator and denominator.
• Using properties of exponents to rewrite \( y^{10} \) as \( y^{5} \) and \( x^{6} \) as \( x^{3} \) after taking their square roots.
• Recognizing that \( \sqrt{9} \) is 3, a basic fact about numbers.

The final result presents a simplified form where complex operations have been broken into more manageable pieces, helping you recognize the underlying relationships between the components.