Problem 5 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(9^{2}\)

Step-by-Step Solution

Answer

The simplified expression of \(9^2\) is 81.

Step 1: Understanding the Expression

The expression to simplify is given as \(9^{2}\). This means 9 is raised to the power of 2, which indicates multiplication of 9 by itself once.

Step 2: Perform the Multiplication

Calculate the value of \(9^{2}\) by multiplying the base, 9, with itself. So, \(9^{2} = 9 \times 9\).

Step 3: Calculate the Result

Multiply 9 by 9, resulting in \(9 \times 9 = 81\).

Step 4: Present the Final Answer

The simplified expression of \(9^{2}\), with positive exponents, is \(81\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra, making complex calculations more manageable. When we simplify an expression like \(9^2\), we are essentially translating it into a form that represents its numeric value directly. To simplify an expression:

• Identify the base and the exponent in the expression.
• Perform the necessary operations as indicated by the exponent.
• Convert the result into its simplest form.

In our example \(9^2\), this involves recognizing that 9 is the base and 2 is the exponent. The operation to be performed is raising 9 to the power of 2, which means multiplying 9 by itself. The product, 81, is the simplified form. Simplification not only makes it easier to understand the value but also prepares us for more complex calculations in algebra.

Positive Exponents

Positive exponents are straightforward to handle since they imply repeated multiplication of the base number. In algebra, an exponent tells us how many times to multiply the base by itself. Having positive exponents, like in \(9^2\), simplifies computations by indicating a clear operation:

• With a positive exponent, you multiply the base by itself as many times as the exponent indicates.
• Positive exponents always yield larger values unless the base is a fraction between 0 and 1.
• The result is always a positive number as long as the base is positive.

So, \(9^2\) tells us to multiply 9, the base, by itself once. This straightforward operation translates the expression into an actual numerical value of 81. By understanding positive exponents, you can simplify expressions accurately and efficiently.

Step-by-Step Solutions

Working through problems with a step-by-step approach is invaluable, especially for equations involving exponents. This methodical approach guides you through each part of the process, ensuring you understand how to arrive at the solution. Let's break down this method:

• Step 1: Identify the Equation - Recognize what the problem is asking you to simplify, such as \(9^2\).
• Step 2: Apply Mathematical Operations - Use the operation indicated by the exponent to perform necessary calculations. For \(9^2\), multiply 9 by 9.
• Step 3: Compute the Result - After performing operations, find the actual value; in this case, 81.
• Step 4: Verify Your Answer - Double-check calculations to ensure accuracy and confirm that the result is in its lowest possible form.

Following these steps ensures not only the correct outcome but a deeper grasp of the problem-solving technique. By consistently using step-by-step solutions, you build a reliable framework for tackling algebraic expressions.