Problem 72 - 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 expression simplifies to \( \frac{343}{125} \).

Step 1: Understand the Negative Exponent

A negative exponent indicates the reciprocal of the base. Therefore, the expression \( \left( \frac{25}{49} \right)^{-3/2} \) becomes \( \left( \frac{49}{25} \right)^{3/2} \).

Step 2: Break Down the Fractional Exponent

The exponent \( \frac{3}{2} \) can be broken down into two operations: raising to the power of 3 (cube) and taking the square root (power of \( \frac{1}{2} \)). Therefore, we'll first take the square root: \( \left( \left( \frac{49}{25} \right)^{1/2} \right)^3 \).

Step 3: Calculate the Square Root

Calculate the square root of the fraction: \( \sqrt{\frac{49}{25}} = \frac{\sqrt{49}}{\sqrt{25}} = \frac{7}{5} \).

Step 4: Raise to the Power of 3

Now, raise \( \frac{7}{5} \) to the power of 3: \( \left( \frac{7}{5} \right)^3 = \frac{7^3}{5^3} = \frac{343}{125} \).

Step 5: Combine Steps and Simplify

So, the simplified expression of \( \left( \frac{25}{49} \right)^{-3/2} \) is \( \frac{343}{125} \).

Key Concepts

Fractional Exponents Explained

When dealing with fractional exponents, it's essential to understand that these exponents represent both roots and powers. Imagine a fractional exponent like \( a^{m/n} \). Here, \( n \) is in the denominator, which indicates the \( n \)th root, and \( m \), the numerator, shows the power to which the base is raised. For example, in \( \left( \frac{25}{49} \right)^{-3/2} \), the exponent \(-3/2\) means two things need to happen:

• The base will first be flipped, due to the negative sign in the exponent.
• It will then be rooted (\( 1/2 \) represents the square root) and powered (raising to the \( 3 \)rd power).

This breakdown means you first handle the root operation and then the power, or do both in any order as each step commutes. Understanding fractional exponents this way helps you simplify expressions involving fractions, roots, and powers with ease.

Simplifying Expressions Step-by-Step

Simplifying expressions requires breaking them down into manageable steps. As seen in simplifying \( \left( \frac{25}{49} \right)^{-3/2} \), you first tackle the negative exponent by taking the reciprocal of the base, turning the expression into \( \left( \frac{49}{25} \right)^{3/2} \).

• First, recognize that the fraction of the fractional exponent involves taking a reciprocal. This flips the original fraction.
• Next, apply the square root operation associated with the \( \frac{1}{2} \) exponent, simplifying the fraction to \( \frac{7}{5} \).
• Finally, raise this result to the power indicated, in this case, cubing the result to further simplify into \( \frac{343}{125} \).

This methodical approach ensures you handle each component of the expression correctly. Simplification involves not just changing the expression's appearance but understanding the operations' sequence to state it in its simplest form.

Understanding Real Numbers

Real numbers include all the numbers on the number line, encompassing both rational and irrational numbers. For our example, the variables represent positive real numbers, which are those greater than zero. This ensures all operations, like square roots, are valid since they're undefined for negative numbers in the realm of real numbers.
Real numbers are important in understanding expressions like \( \frac{49}{25} \), where both numerator and denominator are not just integers but actually positive rational numbers—subsets of real numbers. When working with real numbers in expressions:

• You always assume they're part of the set of numbers that include all rational (fractions, whole numbers) and irrational numbers.
• This assumption is key to handling operations like square roots and fractional exponents.

Thus, being familiar with real numbers helps you confidently navigate expressions, understanding when assumptions such as positivity affect the validity of operations performed.