Problem 1 - Algebra for College Students

Question

For Problems \(1-30\), evaluate each numerical expression. \(81^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

The expression \(81^{\frac{1}{2}}\) evaluates to 9.

Step 1: Identify the Expression

The expression given is \(81^{\frac{1}{2}}\). This represents the square root of 81.

Step 2: Square Root Understanding

The exponent \(\frac{1}{2}\) indicates a square root. Therefore, we need to find a number which, when multiplied by itself, gives 81.

Step 3: Calculate the Square Root

The number that satisfies this condition is 9, since \(9 \times 9 = 81\).

Step 4: Verify the Answer

To verify, we can re-check that \(9^2 = 81\), confirming the answer is correct.

Key Concepts

Numerical Expressions

Numerical expressions are math sentences made up of numbers and operations, such as addition, subtraction, multiplication, and division. They help us express calculations concisely and solve problems efficiently.

In the exercise you see the expression \(81^{\frac{1}{2}}\), which combines numbers with an exponent. When evaluating numerical expressions, it’s essential to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures every calculation is performed correctly.

Here's a simplified example: Given the expression \(8 + 2 \times 3\), you first multiply \(2 \times 3\) to get 6, then add 8 to arrive at 14. This exercise focuses specifically on handling the exponent, an integral part of numerical expressions that we'll delve into next.

Exponents

Exponents are a way to express repeated multiplication of the same number. Written as \(a^b\), the exponent \(b\) tells us how many times the base \(a\) is multiplied by itself.
For example, \(3^2\) means \(3 \times 3\), which equals 9. Exponents simplify expressions and calculations, especially with larger numbers.

In the expression \(81^{\frac{1}{2}}\), the exponent is \(\frac{1}{2}\), indicating a special case called a square root. Understanding exponents is key to mastering numerical expressions, as they frequently appear in various math problems. Remember:

• An exponent of 2 denotes squaring a number.
• An exponent of \(\frac{1}{2}\) points to finding a square root.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. The square root is often denoted by the radical symbol (\(\sqrt{}\)) or as an exponent \(\frac{1}{2}\).
For instance, the square root of 81 is 9, because \(9 \times 9 = 81\). When dealing with square roots, it's important to ensure that your result multiplied by itself truly equals the initial number. This verification is key to checking your work.

How to find square roots efficiently:

• Recognize common perfect squares, such as 1, 4, 9, 16, and so on.
• Estimate using nearby perfect squares if the number isn’t exact.
• Utilize prime factorization for complex numbers.

This concept is vital for algebra and many real-world applications where we need to determine dimensions or solve quadratic equations.