Problem 45 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{25}{16}\right)^{-3 / 2} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{64}{125} \).

Step 1: Understand the Negative Exponent

The expression \( \left(\frac{25}{16}\right)^{-3/2} \) has a negative exponent. By the property of negative exponents, \( x^{-n} = \frac{1}{x^n} \). So, we can rewrite \( \left(\frac{25}{16}\right)^{-3/2} \) as \( \frac{1}{\left(\frac{25}{16}\right)^{3/2}} \).

Step 2: Apply the Power of a Fraction Rule

The next step involves simplifying the inner fraction raised to a power. According to the power of a fraction rule, \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). Thus, \( \left(\frac{25}{16}\right)^{3/2} = \frac{25^{3/2}}{16^{3/2}} \).

Step 3: Calculate the Square Roots

Recognizing that \( a^{3/2} \) is equivalent to \( (a^{1/2})^3 \), calculate the square roots: \( 25^{1/2} = 5 \) and \( 16^{1/2} = 4 \). So we have: \( \frac{25^{3/2}}{16^{3/2}} = \frac{5^3}{4^3} \).

Step 4: Evaluate the Cubes

Find the actual cube of each number: \( 5^3 = 125 \) and \( 4^3 = 64 \). Thus, \( \frac{5^3}{4^3} = \frac{125}{64} \).

Step 5: Reciprocal of the Fraction

Since we started with a negative power, take the reciprocal of the result. So \( \frac{1}{\frac{125}{64}} = \frac{64}{125} \).

Step 6: Finalize the Answer

The expression \( \left(\frac{25}{16}\right)^{-3/2} \) evaluates to \( \frac{64}{125} \). Retain this as the final answer.

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they follow a simple rule. When you see a negative exponent, it tells you to flip the base by taking its reciprocal. In mathematical terms, for any number or expression \(x^{-n}\), this means: \(x^{-n} = \frac{1}{x^n}\). This is exactly what we did in the exercise with \(\left(\frac{25}{16}\right)^{-3/2}\). Instead of immediately jumping to computations, we first expressed the fraction with a positive exponent to make things easier. So, this turned into \(\frac{1}{\left(\frac{25}{16}\right)^{3/2}}\).

• Flipping the base is the key action.
• Always change the negative exponent to positive by taking the reciprocal.

Understanding this concept helps simplify expressions and is a fundamental step toward mastering exponents.

Fractional Exponents

Fractional exponents are another form of notation for roots. When you see something like \(a^{m/n}\), this reflects both an exponent and a root: the base \(a\) is raised to the power \(m\) and then the \(n\)-th root is taken. So it's equivalent to \(\sqrt[n]{a^m}\). In simpler terms, for our specific exercise, \((\frac{25}{16})^{3/2}\) involves both squaring the base and then finding a square root.

• The numerator (3 in our problem) indicates the "power" the base is raised to.
• The denominator (2 in our problem) indicates the type of root to apply.

In the example, to clarify, \(a^{3/2}\) would mean taking the square root first (since 2 is the denominator), and then cubing (as 3 is the numerator), which we applied separately to \(25\) and \(16\).

Power of a Fraction Rule

When dealing with exponents of fractions, it's crucial to apply the power to both the numerator and the denominator of the fraction. The Power of a Fraction Rule tells us that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). Following this rule is crucial to accurately manipulating and simplifying fractions raised to powers.
In our exercise, we had \((\frac{25}{16})^{3/2}\). By applying the power to both, this becomes \(\frac{25^{3/2}}{16^{3/2}}\).

• Make sure to apply the exponent to both parts of the fraction.
• This makes simplifying the expression easier and ensures you maintain balance in the equation.

The consistency of this rule helps break down what might initially seem like complicated expressions.

Cube and Square Roots

Understanding cube and square roots is crucial for simplifying fractional exponents. The language of roots is just another way to express powers in math. A square root asks "what number times itself gives this number?", while a cube root extends this question to "what number times itself three times gives this number?"
In this exercise, when we saw \(25^{3/2}\), it became necessary to first handle \(25^{1/2}\), which is the square root of 25, resulting in 5. The same process was applied to 16, yielding 4. Once these roots were taken, we then cubed both results: \(5^3 = 125\) and \(4^3 = 64\).

• Square roots simplify the base by two-fold repetitive multiplication.
• Cube roots ask for a three-fold multiplication solution.

Once the roots are figured out, the remaining task of exponentiation becomes simpler and manageable, completing the evaluation.