Problem 46 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

\( \left(\frac{16}{9}\right)^{-\frac{3}{2}} = \frac{27}{64} \).

Step 1: Understand the Expression

The given expression is \( \left(\frac{16}{9}\right)^{-\frac{3}{2}} \). This is a power expression which involves both a fraction and a negative exponent. Our task is to simplify it without using a calculator.

Step 2: Address the Negative Exponent

A negative exponent implies taking the reciprocal of the base. Hence, \( \left(\frac{16}{9}\right)^{-\frac{3}{2}} = \left(\frac{9}{16}\right)^{\frac{3}{2}} \).

Step 3: Simplify the Fraction

We now have \( \left(\frac{9}{16}\right)^{\frac{3}{2}} \). This can be broken down as \( \left(\frac{9}{16}\right)^{\frac{1}{2}} \) and then cubed. First, find the square root of the fraction: \( \frac{9}{16} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4} \).

Step 4: Cube the Result

Now, take the result from the previous step, \( \frac{3}{4} \), and raise it to the power of 3: \( \left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64} \).

Step 5: State the Final Result

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

Key Concepts

Negative exponents

Negative exponents can initially seem a bit confusing, but they're easier to understand once you think of them as instructions for inverses. When you see a negative exponent, it tells you to take the reciprocal of the base. For example, if you have an expression like \( a^{-n} \), you simply take the reciprocal of \( a \) and then raise it to the positive power \( n \).

• Step example: \( \left(\frac{16}{9}\right)^{-\frac{3}{2}} = \left(\frac{9}{16}\right)^{\frac{3}{2}} \)
• The base \( \frac{16}{9} \) was inversed to \( \frac{9}{16} \).

This is particularly helpful when simplifying expressions, as it allows you to switch the position of numbers in a fraction, turning division into multiplication. Negative exponents thus streamline calculations and can make mathematical expressions easier to work with once properly understood.

Fractional exponents

Fractional exponents are a way to represent roots using the language of powers. A fractional exponent like \( a^{\frac{m}{n}} \) is a convenient notation that combines the concepts of roots and powers together.

• In the expression \( a^{\frac{m}{n}} \), \( n \) describes the root, and \( m \) describes the power to which the base \( a \) is raised.
• For \( \left(\frac{9}{16}\right)^{\frac{3}{2}} \), you first take the square root (because of the \( \frac{1}{2} \)), resulting in \( \frac{3}{4} \).
• Then, raise the result \( \frac{3}{4} \) to the power of 3.

Using fractional exponents can often make calculations more straightforward, especially when dealing with roots and powers in the same expression, without switching between different types of notation.

Simplifying fractions

Simplifying fractions is a fundamental skill in algebra and involves rewriting a fraction to its simplest form. It sounds complex, but involves straightforward steps:

• First, try to factorize the numerator and the denominator to their prime components if needed.
• Divide the highest common factor out of both the numerator and the denominator.

For fractional exponent expressions, sometimes simplifying involves dealing with roots. For instance,

• In \( \left(\frac{9}{16}\right)^{\frac{1}{2}} = \frac{3}{4} \), the square root of each part (\( \sqrt{9} = 3 \) and \( \sqrt{16} = 4 \)) yields the simplified fraction.

Once the fraction is in its simplest form, performing further operations like exponentiation becomes more manageable. This can often involve repeating the process of simplification until the expression cannot be simplified further.