Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: Express the number in the form \(a / b\), where \(a\) and \(b\) are integers. $$ 16^{-3 / 4} $$

Step-by-Step Solution

Answer

The number is expressed as \(1/8\).

Step 1: Understand the Problem

We are given the expression \(16^{-3/4}\) and need to express it as a fraction, \(a/b\), where \(a\) and \(b\) are integers.

Step 2: Rewrite the Base

Recognize that 16 is a perfect square and can be written as \(16 = 2^4\). This helps in simplifying the expression with fractional exponents.

Step 3: Apply Exponent Rules

Substitute 16 with \(2^4\) in the expression: \((2^4)^{-3/4}\). Then use the property \((x^m)^n = x^{m \cdot n}\) to simplify: \(2^{4\cdot(-3/4)}\).

Step 4: Multiply Exponents

Calculate the exponent: \(4 \cdot (-3/4) = -3\). So, the expression becomes \(2^{-3}\).

Step 5: Express with Positive Exponent

Since the exponent is negative, \(2^{-3}\) can be rewritten as \(1/2^3\) using the rule \(x^{-n} = 1/x^n\).

Step 6: Simplify the Fraction

Calculate \(2^3 = 8\), thus \(1/2^3 = 1/8\). Therefore, the expression \(16^{-3/4}\) is equivalent to \(1/8\).

Key Concepts

Exponent Rules

Exponent rules can be a powerful tool in simplifying mathematical expressions. Let's consider the expression \(16^{-3/4}\). The first step is to recognize that 16 is a power of 2, specifically \(2^4\). This makes it easier to work with when applying exponent rules.

With exponent rules, we use properties that allow us to manipulate exponents systematically:

• \((x^m)^n = x^{m \cdot n}\) allows us to multiply exponents when taking a power of a power.
• A negative exponent, such as \(x^{-n}\), entails taking the reciprocal: \(1/x^n\).

In the case of \(16^{-3/4}\), after rewriting it as \((2^4)^{-3/4}\), applying the rule \((x^m)^n\) simplifies it to \(2^{-3}\). Understanding and applying these exponent rules are essential steps towards simplifying complex expressions.

Fractional Exponents

Fractional exponents may seem challenging, but they are just another way to express roots and powers. An expression such as \(16^{-3/4}\) combines both a fraction and a negative exponent.

A fractional exponent, \(a^{m/n}\), actually means \(a\) raised to a power \(m\) and then taking the \(n\)-th root. For instance, \(16^{1/4}\) represents the fourth root of 16. Similarly, \(16^{3/4}\) means the fourth root of 16 cubed. In the given expression, \(-3/4\) implies a combination of taking roots and the reciprocal, due to the negative sign.

Decoding fractional exponents requires practice:

• Positive fractional exponents indicate roots, i.e., \(a^{1/n} = \sqrt[n]{a}\).
• If there's a negative sign involved, it results in taking the reciprocal of the root.

Breaking down fractional exponents makes handling even complex expressions manageable.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. With expressions like \(16^{-3/4}\), simplification processes use a combination of exponent rules and other math principles.

The expression can initially look complicated, but can become manageable with systematic steps. Starting with rewriting bases, such as converting 16 to \(2^4\), sets a foundation for simplification. Employing exponent rules to tackle the fractional exponent transforms it to \(2^{-3}\).

The simplification of \(2^{-3}\) into a fraction, \(1/2^3\), and then calculating, \(1/8\), is the final step:

• Identify and transform bases for simplification.
• Apply exponent rules to simplify expressions with fractional exponents.
• Calculate exact values to express the simplest form.

By methodically breaking down the expression, simplifying becomes straightforward, transforming complex equations into simple fractions or numbers.