Problem 43 - Algebra A Combined Function

Question

Evaluate each exponential expression. $$ 25^{1 / 2} $$

Step-by-Step Solution

Answer

The expression evaluates to 5.

Step 1: Understanding the Exponent

The expression given is \( 25^{1/2} \). The exponent \( 1/2 \) means to take the square root of the base number 25. Essentially, \( x^{1/2} \) is equivalent to \( \sqrt{x} \).

Step 2: Calculating the Square Root

To evaluate \( 25^{1/2} \), you need to find the number which, when multiplied by itself, equals 25. This number is 5, as \( 5 \times 5 = 25 \). Therefore, \( \sqrt{25} = 5 \).

Step 3: Final Evaluation

Since the square root of 25 is 5, this means that \( 25^{1/2} = 5 \). We have evaluated the exponential expression by finding the square root.

Key Concepts

Square Root

A square root finds the number that, when multiplied by itself, gives the original number. For example, finding the square root of 25 is asking, "What number, when squared, equals 25?" To put it simply, if you multiply 5 by itself, you get 25. Therefore, the square root of 25 is 5.

The square root is denoted by the radical symbol "\( \sqrt{} \)". If you see \( \sqrt{25} \), it means you're looking for the square root of 25, which is 5.

• \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \)
• \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \)
• \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \)

Understanding square roots is fundamental in solving various mathematical problems, especially when they're represented as exponential expressions.

Evaluating Expressions

Evaluating expressions involves solving or simplifying mathematical expressions to find their value. It requires following certain rules and operations to reach the final answer.

When dealing with expressions like \( 25^{1/2} \), we evaluate them by following specific steps:

• Identify and understand the components: In our example, 25 is the base, and \( \frac{1}{2} \) is the exponent which indicates the operation to perform.
• Use mathematical operations appropriately: Here the exponent \( \frac{1}{2} \) indicates taking the square root of 25.
• Calculate the result carefully: Determine the value of the expression, which in this case is 5 since \( \sqrt{25} = 5 \).

Evaluating expressions is about simplifying them step-by-step, ensuring that each component and operation is understood and correctly applied. This process helps students gain clarity and confidence in their math skills.

Rational Exponents

Rational exponents are expressions where the exponent is a fraction, like \( x^{\frac{m}{n}} \). These exponents indicate both power and root calculations, adding flexibility to how we represent and solve problems.

The expression \( x^{\frac{1}{2}} \) is a common example of a rational exponent, and it signifies the square root of \( x \). Thus, \( 25^{\frac{1}{2}} \) tells us to find \( \sqrt{25} \), which is 5. Similarly, any expression of the form \( x^{\frac{1}{n}} \) represents the \( n \)-th root of \( x \).

• \( 64^{\frac{1}{3}} \) represents \( \sqrt[3]{64} \) which equals 4, since \( 4 \times 4 \times 4 = 64 \).
• \( 81^{\frac{1}{4}} \) means \( \sqrt[4]{81} \), which is 3 because \( 3 \times 3 \times 3 \times 3 = 81 \).

Understanding rational exponents is crucial as it connects the concept of roots with powers, allowing for a richer understanding and more diverse problem-solving techniques in mathematics.