Problem 37 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ \frac{\left(2(3)^{2}+\frac{1}{3^{-2}}\right)^{\frac{2}{3}}}{6\left(2+\frac{1}{4}\right)^{-\frac{1}{2}}} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \(\frac{9}{4}\).

Step 1: Calculate inside the numerator

Start by simplifying the expression inside the numerator, \( 2(3)^2 + \frac{1}{3^{-2}} \). Calculate \( (3)^2 \) to get \( 9 \). So, the term becomes \( 2 \times 9 + \frac{1}{3^{-2}} \).

Step 2: Simplify \( \frac{1}{3^{-2}} \)

The expression \( \frac{1}{3^{-2}} \) can be rewritten as \( (3^2) \) because \( x^{-a} = \frac{1}{x^a} \). Therefore, \( \frac{1}{3^{-2}} = 9 \).

Step 3: Simplify the numerator

Now, simplify the expression \(2 \times 9 + 9\), which becomes \(18 + 9 = 27\).

Step 4: Calculate the power in the numerator

Apply the power \(\frac{2}{3}\) to \(27\). Since \(27 = 3^3\), we raise it to \(\frac{2}{3}\), obtaining \((3^3)^{\frac{2}{3}} = 3^2 = 9\).

Step 5: Simplify inside the denominator

For the denominator, simplify \(2 + \frac{1}{4}\). Convert \(2\) into a fraction \(\frac{8}{4}\) and add to \(\frac{1}{4}\) to get \(\frac{9}{4}\).

Step 6: Apply the negative power in the denominator

Raise \(\frac{9}{4}\) to the power of \(-\frac{1}{2}\). It becomes \((\frac{9}{4})^{-\frac{1}{2}} = (\frac{4}{9})^{\frac{1}{2}} = \frac{2}{3}\).

Step 7: Evaluate the full expression

Now calculate the expression: \[\frac{9}{6 \times \left( \frac{2}{3} \right)} = \frac{9}{6 \times \frac{2}{3}} = \frac{9}{4}.\]

Step 8: Simplify the final result

Simplifying \(\frac{9}{4}\) gives us the rational number in its simplest form, since it cannot be reduced further.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. The base is the number that is being multiplied by itself, while the exponent indicates how many times the base is used as a factor. For instance, in the expression \(3^2\), 3 is the base and 2 is the exponent. That means 3 is multiplied by itself once (i.e., \(3 \times 3\)). This results in 9. Similarly, any time you see a number raised to a power, it means repeatedly multiplying the number by itself.
When working with expressions like our original exercise, understanding exponent rules is essential. Notice how \((3^3)^\frac{2}{3}\) simplifies to \(3^2\), using the property \((x^m)^n = x^{m \cdot n}\). Exponentiation is powerful because it allows for compact representation of repeated multiplication.

Fraction Simplification

Fraction simplification is all about making a fraction as simple as possible. A fraction is simplified when both its numerator and its denominator have no common factors (besides 1). For example, \(\frac{9}{6}\) can be simplified. The greatest common divisor of 9 and 6 is 3. Dividing both the numerator and denominator by 3 gives \(\frac{3}{2}\).
In more complex expressions, like in our exercise, break down each part one step at a time. First, ensure individual fractions such as \(2 + \frac{1}{4}\) are expressed as a single fraction before applying operations like exponentiation. Be diligent and work through each term carefully to achieve the simplest form for each element involved.

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the positive exponent. In simple terms, \(x^{-a} = \frac{1}{x^a}\). This is a crucial concept and often comes into play in expressions that need to be simplified.
Consider \(3^{-2}\) as found in our exercise. It translates to \(\frac{1}{3^2}\), which equals \(\frac{1}{9}\). Understanding how to handle negative exponents helps avoid common pitfalls when simplifying expressions. It turns apparent division into multiplication of reciprocals, streamlining complex calculations. Always remember to convert negative exponents into fractional form to make arithmetic easier.

Power of a Number

The power of a number describes how many times to use the number in a multiplication. When a number is raised to a power (also known as an exponent), it reflects repeated multiplication. For practical computation, recognize patterns such as \((b^m)^n = b^{m \cdot n}\), used when simplifying powers.
In the exercise, we applied this rule: \((3^3)^\frac{2}{3} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9\). Here, knowing 27 as \(3^3\) was key to simplifying using its power. Understanding how to apply and simplify power operations can resolve complex expressions into manageable numbers, simplifying your overall workload and yielding precise results.