Problem 113 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ 16^{-3 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{64} \).

Step 1: Write the Expression in Radical Form

The given expression is \( 16^{-3/2} \). To simplify, first rewrite it in radical form. Remember that if a term is in the form \( x^{-a/b} \), it can be rewritten as \( \frac{1}{x^{a/b}} \), which equals \( \frac{1}{(x^b)^a} \). So, \( 16^{-3/2} \) becomes:\[ \frac{1}{16^{3/2}} \]

Step 2: Rewrite the Denominator as a Power

We need to simplify \( 16^{3/2} \). Start by rewriting it using the property \( (x^m)^n = x^{m \cdot n} \). So:\[ 16^{3/2} = (16^{1/2})^3 \]

Step 3: Evaluate the Square Root

Calculate \( 16^{1/2} \). Since the square root of 16 is 4, we have:\( 16^{1/2} = 4 \)Thus, the expression becomes:\[ (4)^3 \]

Step 4: Calculate the Power of 4

Raise 4 to the power of 3:\( 4^3 = 4 \times 4 \times 4 = 64 \)So, \( 16^{3/2} = 64 \).

Step 5: Simplify the Entire Fraction

Substitute back into the expression:\[ \frac{1}{16^{3/2}} = \frac{1}{64} \]So, the simplified expression is \( \frac{1}{64} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. They are crucial in simplifying expressions where powers or exponents are involved. For instance, to simplify the expression of a form like \( x^{a/b} \), you break it down into radicals. Applying this to our exercise, we had \( 16^{-3/2} \).

• The root, \( \frac{1}{2} \), tells us to find the square root.
• The exponent, \( 3 \), is what we will eventually raise the result to.

By understanding how to manipulate radical expressions, you can easily move between exponential and radical forms to simplify expressions efficiently.

Exponents

Exponents are a shorthand notation for repeated multiplication of a number by itself. They play a significant role in algebra, especially when working with powers. In our example, we start with \( 16^{-3/2} \):

• The negative sign indicates a reciprocal.
• The fraction \( \frac{3}{2} \) facilitates translating between exponentials and roots.

Exponents make calculations involving large numbers manageable and help in expressing any number as a power of its factors. They are foundational for simplifying complex algebraic expressions and solving equations.

Simplifying Expressions

Simplifying expressions is about transforming them into the simplest or most reduced form possible. It's an essential algebraic skill that involves reducing the complexity of the expression while preserving its value. In our exercise, we simplified \( 16^{-3/2} \) through several steps:

• Convert exponents to a radical form \((\frac{1}{16^{3/2}})\).
• Solve the root \((16^{1/2} = 4)\).
• Compute the power \((4^3)\).
• Finally, simplify \(\frac{1}{64}\).

Each step ensures the expression is easier to handle and calculate, revealing the underlying mathematical properties. Simplification aids in understanding and applying further algebraic transformations and problem-solving.