Problem 46 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{16}{9}\right)^{-3 / 2} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{27}{64} \).

Step 1: Understand the Negative Exponent

The expression \( \left(\frac{16}{9}\right)^{-3/2} \) has a negative exponent. When we have a negative exponent, we take the reciprocal of the base. This gives us: \( \left(\frac{9}{16}\right)^{3/2} \).

Step 2: Convert the Exponent to a Radical

The exponent \( \frac{3}{2} \) can be rewritten in terms of a square root and a cube. Specifically, \( \left(\frac{9}{16}\right)^{3/2} \) is equivalent to \( \left(\left(\frac{9}{16}\right)^{1/2}\right)^3 \).

Step 3: Calculate the Square Root

Calculate \( \left(\frac{9}{16}\right)^{1/2} \), which is the square root of \( \frac{9}{16} \). The square root of 9 is 3, and the square root of 16 is 4, so \( \left(\frac{9}{16}\right)^{1/2} = \frac{3}{4} \).

Step 4: Cube the Result

Now, take the result from Step 3 and cube it: \((\frac{3}{4})^3\). Cubing the fraction gives \( \frac{3^3}{4^3} = \frac{27}{64} \).

Step 5: Final Answer

The value of \( \left(\frac{16}{9}\right)^{-3/2} \) is \( \frac{27}{64} \).

Key Concepts

Negative Exponents

Negative exponents might seem a bit confusing at first, but they aren't as intimidating as they appear. When you encounter a negative exponent, it essentially suggests taking the reciprocal of the base. This changes the problem from division-oriented to multiplication-oriented.

For example, with the expression \( a^{-b} \), you would rewrite it as \( \frac{1}{a^b} \). By taking the reciprocal, the negative sign in the exponent disappears.

Another way to think of this is moving the base to the other side of the fraction. In the exercise, the negative exponent in \( \left(\frac{16}{9}\right)^{-3/2} \) prompts us to switch the fraction to \( \left(\frac{9}{16}\right)^{3/2} \). This transformation simplifies the problem and paves the way for the next calculations.

Fractional Exponents

Fractional exponents can be viewed as a way to combine both roots and powers into a single operation. The numerator of the fractional exponent is the power, and the denominator is the root.

Think of it: in \( x^{m/n} \), \( n \) represents the root you should take first, and \( m \) tells you the power to raise that result to.

For the exercise expression \( \left(\frac{9}{16}\right)^{3/2} \), the 2 in the denominator signifies taking the square root first. Only after obtaining this result do you raise it to the 3rd power, as indicated by the numerator.

The great thing about fractional exponents is that they allow you to streamline complex operations involving roots and powers into a tidy, manageable form.

Rational Exponents

Rational exponents might sound tricky, but they’re simply another way to express roots and powers further simplified using fractional exponents. Rational exponents, like \( x^{1/n} \), indicate the \( n \)th root of \( x \). This is similar to fractional exponents but can sometimes be even more straightforward.

In rational exponent terms, \( x^{3/2} \) means you take the square root (because of \( 1/2 \)), and then cube the result (because of the numerator, 3). This provides a powerful tool for managing more complicated expressions.

When we apply rational exponents to the given exercise by breaking down \( (9/16)^{3/2} \), it shows how a foundational understanding of these exponents enables you to solve expressions that might otherwise seem unapproachable, all without a calculator's assistant.