Problem 75 - Intermediate Algebra

Question

$$ \frac{1}{9^{-5 / 2}} $$

Step-by-Step Solution

Answer

The expression simplifies to 243.

Step 1: Understand Negative Exponent

Start by understanding that a negative exponent indicates that we should take the reciprocal of the base. For example, a base of \(9^{-5/2}\) becomes \(\frac{1}{9^{5/2}}\).

Step 2: Apply Reciprocal to the Fraction

Since we have \(\frac{1}{9^{-5/2}}\), and \(-5/2\) is the exponent, the negative exponent tells us to take the reciprocal. This gives us \(9^{5/2}\).

Step 3: Simplify the Exponentiation

The expression \(9^{5/2}\) can be rewritten using radicals as: \((9^{1/2})^5 = (\sqrt{9})^5\). Since \(\sqrt{9} = 3\), we now have \(3^5\).

Step 4: Calculate the Final Answer

Compute \(3^5\) by multiplying: \(3 \times 3 = 9\), \(9 \times 3 = 27\), \(27 \times 3 = 81\), and finally \(81 \times 3 = 243\). Therefore, \(9^{5/2} = 243\).

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is vital, especially when dealing with negative exponents. The reciprocal of a fraction is achieved by swapping its numerator and denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
This concept is helpful when handling negative exponents since a negative exponent indicates taking the reciprocal of the base.
When you see an expression like \(9^{-5/2}\), it suggests that you first take the reciprocal of the base, transforming it from \(9^{-5/2}\) to \(\frac{1}{9^{5/2}}\).

• The original fraction \(\frac{1}{9^{-5/2}}\) becomes the new base after taking the reciprocal, which flips the exponent to a positive, effectively rewriting it as \(9^{5/2}\).

This process allows us to work with expressions that may initially appear complex or challenging by simplifying them through reciprocation.

Radicals and Exponents

Radicals and exponents are closely tied to each other and understanding them can simplify expressions considerably. An exponent like \(^{1/2}\) denotes a square root. Thus, \(a^{1/2} = \sqrt{a}\).
When an exponent is expressed as a fraction, it represents a radical. The denominator becomes the type of root, and the numerator stays as the power.

• For example, \(9^{5/2}\) can be broken down into \((9^{1/2})^5\), meaning that you first take the square root of 9, which is 3, and then raise it to the power of 5.

Using radicals simplifies recognizing and calculating expressions involving fractional exponents, making it a handy technique when tackling problems.
By converting to radicals and simplifying, we can often reduce calculation complexity.

Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible and involves several useful techniques. For instance, turning a complex expression like \(9^{5/2}\) into a simpler form through step-by-step breakdown.

• First, recognize that \(9^{5/2} = (9^{1/2})^5\), and replace \(9^{1/2}\) with its simpler form, \(\sqrt{9} = 3\).
• Then, compute \(3^5\) to get 243.

Simplifying deeply focuses not only on making calculations manageable but also on improving clarity and understanding.
Utilizing properties of exponents and radicals can significantly reduce the complexity of expressions, making it essential for efficient problem-solving.
Breaking down each part using various algebraic identities can often make the task more intuitive.