Problem 72 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$\left(\frac{81 y^{16}}{16 x^{12}}\right)^{\frac{1}{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{9y^8}{4x^6} \)

Step 1: Identify the Exponents

We will start by identifying the exponents for each of the numerical values and variables within the brackets: \[ \left( \frac{81 y^{16}}{16 x^{12}} \right)^{\frac{1}{2}} \]. Here, we can see that 81, \(y^{16}\), 16 and \(x^{12}\) are all raised to the power of \(\frac{1}{2}\).

Step 2: Simplify the Numerical Part

Next, simplify the root of the numerical values \(81^\frac{1}{2}\) and \(16^\frac{1}{2}\). The square root of 81 is 9, and the square root of 16 is 4. Therefore, the fraction becomes \(\frac{9}{4}\)

Step 3: Simplify the Variable Part

Now, simplify the variables \(y^{16}\) and \(x^{12}\) both raised to the power \(\frac{1}{2}\). Following the rule of exponents that states \( (a^n)^m = a^{n*m}\), the variable parts become \(y^{16*\frac{1}{2}}\) and \(x^{12*\frac{1}{2}}\), which simplifies to \(y^8\) and \(x^6\). So, the expression is now \(\frac{9y^8}{4x^6}\)

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical expressions into a more manageable form. This process is essential in solving equations and understanding mathematical relationships.
Let's simplify the expression step-by-step. First, identify the components within the expression that can be combined or reduced.

• Numerical values: Simplify any numbers first. Look out for integers that can be factored or broken into their prime components.
• Variables: Pay attention to the parts involving variables, and apply appropriate exponent rules.

For example, take the expression: \[\left(\frac{81 y^{16}}{16 x^{12}}\right)^{\frac{1}{2}}.\]Starting with numerical simplification, 81 is reduced to 9 with its square root, and 16 to 4.
Next, simplify the variables using exponent rules, which we'll discuss in more detail. The resulting expression becomes: \[\frac{9y^8}{4x^6}.\]
Remember, simplification often helps to better understand and solve problems by converting them into their simplest forms.

Square Roots

Square roots are fundamental concepts in mathematics, used to find a number which, when multiplied by itself, results in the original number.
For instance, the square root of 81 is 9 because 9 multiplied by 9 equals 81.
Understanding how square roots work allows us to simplify expressions easily. Let's consider the square root aspect in more detail.

• Prime Factorization: Break down the number into its prime factors, which helps in identifying its square roots. For example, 16 is the same as \(2^4\).
• Perfect Squares: A number whose square root results in an integer. Both 81 and 16 are perfect squares, which simplifies the calculation to 9 and 4, respectively.

In expressions, identifying perfect squares forms the foundation for simplification, especially when fractional exponents are in play. This is crucial while working within brackets of an expression that needs simplifications, like in \[\left(\frac{81 y^{16}}{16 x^{12}}\right)^{\frac{1}{2}}.\] Mastering square roots is an indispensable tool for simplifying mathematical expressions.

Fractional Exponents

Fractional exponents may appear daunting, but they are just an extension of the idea of square roots. A fractional exponent like \(\frac{1}{2}\) represents a root—specifically the square root if it's \(\frac{1}{2}\).
For instance, any number \(a\) raised to the power of \(\frac{1}{2}\) is equivalent to the square root of \(a\), or \(\sqrt{a}.\) In our expression, \[\left(\frac{81 y^{16}}{16 x^{12}}\right)^{\frac{1}{2}},\]let's break down how this works:

• Applying Fractional Exponents to Numbers: \(81^{\frac{1}{2}}\) becomes 9 and \(16^{\frac{1}{2}}\) becomes 4.
• Applying Fractional Exponents to Variables: Use the rule \((a^n)^m = a^{n*m}\), so \(y^{16}\) raised to \(\frac{1}{2}\) becomes \(y^{8}\), and \(x^{12}\) becomes \(x^{6}\).

Dealing with fractional exponents helps convert complex expressions into simpler forms. Each root calculation breaks down the problem into manageable components, facilitating easier handling and better understanding of an expression. Such simplifications are practical in solving equations efficiently and accurately.