Question

Evaluate the expression without using a calculator. $$ 4^{5 / 2} $$

Short answer

The expression \(4^{5 / 2}\) evaluates to 32.

Step-by-step solution

Understanding Fractional Exponents

The expression \(4^{5 / 2}\) can be re-written using the fractional exponent as \(\sqrt{4^5}\), since the denominator '2' in the fractional exponent represents the square root and the numerator '5' is the power to which 4 needs to be raised.

Applying the Exponent Part

Raising 4 to the power of 5 gives \(4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024\). So, \(\sqrt{4^5}\) becomes \(\sqrt{1024}\).

Taking the Square Root

The square root of 1024 is 32. Therefore, \(\sqrt{1024} = 32\).

Key concepts

Square Root

The square root of a number is an operation that finds a value which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. The symbol for square root is \( \sqrt{} \), and it specifically refers to the "+" value rather than negatives. Finding square roots can be seen as reversing the process of squaring a number. When dealing with fractional exponents like in the exercise \(4^{5/2}\), the square root is used because the denominator of the exponent is "2".

• When you see a fractional exponent of \(\frac{1}{2}\), think square root.
• It is one of the most used roots in mathematics due to its simplicity and practical applications.

Splitting the exponent into a numerator and denominator helps to first raise the base to a power, followed by taking the square root, ensuring accuracy and simplicity in solving these expressions.

Power and Exponents

In mathematics, powers and exponents are the way we express repeated multiplication of a number by itself. The base number is what you are multiplying, and the exponent indicates how many times you multiply the base by itself. For the expression \(4^{5/2}\), the base 4 is raised to the power of 5, and the resulting value is 1024.

• Raising a number to a power is simply multiplying the base by itself as many times as the exponent indicates.
• For powers such as 5, you multiply the base number, here 4, five times: \(4 \times 4 \times 4 \times 4 \times 4\).

Using exponents allows for simplifying expressions and calculations, especially when dealing with large numbers or multiple repeated multiplications. Understanding how to manipulate these powers through laws of exponents also aids in simplifying expressions such as those with fractional powers.

Evaluating Expressions

Evaluating expressions involves performing the operations according to the given mathematical rules to find a numerical value for the expression. In the given exercise, it's important to understand how to handle both the exponent and the square root components.

• First, use the laws of exponents to simplify the power operation: \(4^5\).
• Next, transform \(4^{5/2}\) into a format you can calculate: \(\sqrt{4^5}\).
• Finally, follow through by calculating and simplifying the square root \(\sqrt{1024} = 32\).

Evaluating expressions is a step-by-step process that requires applying algebraic rules. It includes switching from exponential to root operations, ensuring each step logically follows the next. Mastery in evaluating expressions with different types of exponents, like fractional ones, allows one to solve complex problems with ease and accuracy.