Problem 72 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ \left(\frac{25}{49}\right)^{-3 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{343}{125}\).

Step 1: Understand the Negative Exponent Rule

Recall that to deal with a negative exponent, we use the rule \(a^{-n} = \frac{1}{a^{n}}\). Here, \(a = \frac{25}{49}\) and \(n = \frac{3}{2}\).

Step 2: Apply the Negative Exponent

Apply the negative exponent rule to convert the expression: \(\left(\frac{25}{49}\right)^{-\frac{3}{2}} = \frac{1}{\left(\frac{25}{49}\right)^{\frac{3}{2}}}\).

Step 3: Simplify the Positive Exponent

Now simplify \(\left(\frac{25}{49}\right)^{\frac{3}{2}}\). This means finding the square root and then cubing: \(\left(\frac{25}{49}\right)^{\frac{3}{2}} = \left(\sqrt{\frac{25}{49}}\right)^{3}\).

Step 4: Calculate the Square Root

The square root of a fraction is found by taking the square root of the numerator and the square root of the denominator separately: \(\sqrt{\frac{25}{49}} = \frac{\sqrt{25}}{\sqrt{49}} = \frac{5}{7}\).

Step 5: Cube the Result

Cube the fraction obtained: \(\left(\frac{5}{7}\right)^{3} = \frac{5^3}{7^3} = \frac{125}{343}\).

Step 6: Compute the Reciprocal

Since the expression due to the negative exponent was initially \(\frac{1}{\left(\frac{25}{49}\right)^{\frac{3}{2}}}\), take the reciprocal of \(\frac{125}{343}\), giving us \(\frac{343}{125}\).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when simplifying expressions with exponents, especially when dealing with negative exponents. The key rules to remember include:

• Negative Exponent Rule: Use the formula \(a^{-n} = \frac{1}{a^{n}}\). This means a negative exponent signifies the reciprocal of the base raised to the positive form of the exponent.
• Power of a Power: When raising a power to a power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Base Raising: With fractional bases like \(\left(\frac{25}{49}\right)^{-\frac{3}{2}}\), remember both components (numerators and denominators) participate in exponentiation.

These rules allow us to transform seemingly complex expressions into more manageable forms, especially when negative exponents are involved.

Fraction Exponentiation

Fraction exponentiation might appear complex at first, but breaking it down simplifies the process. A fractional exponent indicates two operations: root extraction and raising to a power. For instance, \(\left(\frac{25}{49}\right)^{\frac{3}{2}}\) means:

• Step 1: Take the square root (\(\frac{1}{2}\) as an exponent) of both numerator and denominator separately: \(\sqrt{\frac{25}{49}} = \frac{5}{7}\).
• Step 2: Then, raise the result to the third power (cubing): \(\left(\frac{5}{7}\right)^3 = \frac{125}{343}\).

By combining roots and powers, fractional exponents allow for powerful yet concise expression simplification, even if they initially seem intimidating.

Simplifying Fractions

Simplifying fractions is a process of turning fraction expressions into their simplest form. During our exercise, simplifying involved:

• Determining the square root of both the numerator and the denominator separately: \(\sqrt{25} = 5\) and \(\sqrt{49} = 7\).
• Cubing the simplified fraction: \(\left(\frac{5}{7}\right)^3 = \frac{125}{343}\).

Once expressions are evaluated fully, you might need to further simplify by ensuring the fraction is reduced to its lowest terms, though in this example, \(\frac{343}{125}\) cannot be simplified further. Understanding the interplay between exponent rules and fraction operations is key to simplifying efficiently.