Problem 70 - Algebra 1

Question

Evaluate the expression. $$a^{4}+8 \text { when } a=10$$

Step-by-Step Solution

Answer

The value of the expression \(a^{4}+8\) when \(a=10\) is 10,008.

Step 1: Substitute the value of a into the expression

Replace \(a\) with 10 in the expression. This gives: \(10^{4}+8\)

Step 2: Perform exponentiation

Calculate the value of \(10^{4}\), which is 10,000.

Step 3: Perform the addition

Add 8 to 10,000, which equals 10,008.

Key Concepts

Substitution

In mathematics, substitution is a method used to replace a variable with a specific value. This is the first step in evaluating expressions involving variables. For instance, when we encounter the expression \(a^4 + 8\) and are told that \(a = 10\), we simply replace every occurrence of \(a\) with 10. This transforms our expression into \(10^4 + 8\).

Substitution is a critical skill as it allows students to handle expressions that include variables and make them more concrete. It's akin to replacing a blank with a number in a math problem. Make sure to substitute carefully, as any errors in this step will affect the entire calculation. After substitution, our abstract expression becomes ready for further operations like exponentiation and addition.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. This process is like repeated multiplication. For example, in the expression \(10^4\), the number 10 is raised to the power 4, meaning 10 is multiplied by itself four times:

\(10^4 = 10 \times 10 \times 10 \times 10 = 10,000\).

This operation is often represented by writing the base number followed by a small number (the exponent) to the top right, which shows how many times to multiply the base by itself. Exponentiation simplifies complex multiplication and is an essential concept for many areas of math and science. Understanding exponentiation helps students handle large calculations more easily and appreciate its application in real-world scenarios, like computing squares and cubes.

Addition

Addition is one of the basic arithmetic operations, which involves combining two or more numbers to get a total. In our evaluated expression, after completing the exponentiation step and arriving at \(10,000\), the next step is to add 8.

This process looks like:

• Start with the result from exponentiation, 10,000.
• Add the constant: \(10,000 + 8\).
• Write down the total: 10,008.

Addition is straightforward and follows the rule of bringing together quantities. It's important to ensure accuracy in addition, as small books can lead to unexpectedly large errors, especially in a series of calculations. Mastering addition is fundamental, not only because it is a primary operation but also because it forms the basis of more complex arithmetic operations.