Problem 20 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 16^{3 / 2} $$

Step-by-Step Solution

Answer

The value of \( 16^{3/2} \) is 64.

Step 1: Understand Exponent Notation

The exponent notation \( 16^{3/2} \) indicates that 16 is raised to the power of \( \frac{3}{2} \). This is equivalent to taking the square root of 16 first (because of the denominator 2) and then cubing the result (because of the numerator 3).

Step 2: Calculating the Square Root

Calculate the square root of 16. Since \( 16 = 4^2 \), the square root of 16 is 4, thus \( \sqrt{16} = 4 \).

Step 3: Raise the Result to the Power of 3

Now we take the result from Step 2, which is 4, and raise it to the power of 3. Calculate \( 4^3 = 4 \times 4 \times 4 \). First, \( 4 \times 4 = 16 \), and then \( 16 \times 4 = 64 \).

Step 4: Final Expression Evaluation

Combine all steps to evaluate the expression: \( 16^{3/2} = (\sqrt{16})^3 = 4^3 = 64 \). We have calculated that \( 16^{3/2} = 64 \).

Key Concepts

Understanding Square Roots

The square root is a mathematical operation that helps us find a number which, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4. This is because 4 multiplied by 4 equals 16.

• Square roots are usually indicated with the radical symbol: \( \sqrt{} \).
• The square root of a number \( x \) gives a result \( y \), where \( y \times y = x \).

Calculating square roots is a fundamental skill in algebra. Understanding this helps when dealing with more complex expressions, such as those involving fractional exponents. You will often start by simplifying expressions using square roots before proceeding to other operations. In our example, the square root of 16 simplifies to 4, which becomes the base for further calculations.

Fractional Exponents Explained

Fractional exponents, like \( 16^{3/2} \), provide a way to express roots and powers in a single notation. The expression \( a^{m/n} \) means you take the \( n^{th} \) root of \( a \) and then raise the result to the \( m^{th} \) power.

• The denominator tells you which root to take. For example, \( 16^{3/2} \) involves taking the square root because the denominator is 2.
• The numerator tells you the power to raise the root to, so you then cube the result in this example.

Fractional exponents are a powerful tool in algebra because they allow for the combination of roots and exponents in a single, simplified expression. They extend the concept of powers beyond whole numbers, allowing for nuanced calculations as seen in our example where \( 16^{3/2} \) simplifies using both square roots and cubing.

Algebraic Concepts in Simplification

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It is fundamental in solving equations and simplifying expressions like those with fractional exponents. When simplifying \( 16^{3/2} \), we use algebraic principles such as the rules of exponents and roots:

• Breaking down the expression into manageable parts using exponent rules.
• Understanding how to apply operations like square roots and powers sequentially.

In our exercise, the algebraic process involved simplifying the expression by first considering the square root, then applying the exponent. This two-step approach is a hallmark of algebraic thinking, optimizing the use of mathematical rules to reach a solution efficiently. Algebra not only simplifies these computations but also helps in understanding connections between different mathematical operations.