Problem 19 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 25^{3 / 2} $$

Step-by-Step Solution

Answer

The value of \(25^{3/2}\) is 125.

Step 1: Understand the Exponent

The expression \(25^{3/2}\) involves a fractional exponent. The fraction \(\frac{3}{2}\) signifies two operations: squaring and taking a square root. If you split the exponent as \(25^{\frac{1}{2} \times 3} = (25^{\frac{1}{2}})^3\), it means we first find the square root and then cube the result.

Step 2: Calculate the Square Root

Find \(25^{\frac{1}{2}}\), which is the square root of 25. Since \(25 = 5 \times 5\), the square root of 25 is 5. Therefore, \(25^{\frac{1}{2}} = 5\).

Step 3: Cube the Result

Now, use the result from Step 2 to find \((25^{\frac{1}{2}})^3 = 5^3\). To cube 5, multiply it by itself three times: \(5 \times 5 \times 5 = 125\).

Step 4: Verify the Simplified Process

Alternatively, reverse the order of operations. Compute \(25^3\) first as \(15625\) (25 multiplied three times), then find \((15625)^{\frac{1}{2}}\). However, for practical simplicity and assurance, trust the initial method.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is raised to a power or exponent. The exponent indicates how many times the base is multiplied by itself. For fractional exponents like those in this exercise, the process involves additional operations.
When dealing with a fractional exponent, such as \( \frac{3}{2} \), it represents a combination of two operations: a root and a power. The denominator of the exponent (here, 2) denotes the type of root, while the numerator (here, 3) indicates the power to which the base is raised after the root is applied.

• The base 25 raised to the power of \( \frac{3}{2} \) involves first finding the square root (due to the denominator of 2) and then cubing the result (due to the numerator of 3).
• This is equivalent to writing \( 25^{\frac{3}{2}} = (25^{\frac{1}{2}})^3 \).
• Understanding this split helps simplify complex fractional exponent calculations.

Square Root

The square root of a number is a value which, when multiplied by itself, gives the original number. In mathematical notation, finding the square root is expressed as raising a number to the power of \( \frac{1}{2} \). For example, the square root of 25 is 5, since \( 5 \times 5 = 25 \).

• The square root function is a key component in formulas with fractional exponents, where the exponent's denominator indicates the required root.
• For the exercise, \( 25^{\frac{1}{2}} \) was calculated as 5, simplifying further operations.
• It is essential to identify and correctly apply this operation to handle expressions involving roots and powers successfully.

Mathematical Operations

Mathematical operations are essential processes like addition, subtraction, multiplication, and division, used to solve equations. In the context of our exercise, they also include exponentiation and root extraction.
When performing mathematical operations with fractional exponents, it's important to approach them step-by-step:

• Identify the operations needed from the exponent: root extraction from the denominator and raising to the power from the numerator.
• Carefully perform each operation in sequence to avoid mistakes, beginning with simpler calculations like roots.
• Verify results by considering alternative solving methods or by checking with the reverse order, ensuring the accuracy of complex expressions.

Practicing these steps helps in confidently managing more complicated mathematical expressions.