Problem 7 - Beginning and Intermediate Algebra

Question

The value of \(5 x^{2}\) for \(x=4\) was found incorrectly as follows. $$ \begin{array}{l} 5 x^{2} \\ =5 \cdot 4^{2} \\ =20^{2} \\ =400 \end{array} $$ Find the correct value of the expression.

Step-by-Step Solution

Answer

The correct value is 80.

Step 1: Understand the expression

The given expression is \(5 x^2\) and we need to find its value for \(x = 4\).

Step 2: Substitute the value of x

Substitute \(x = 4\) into the expression \(5 x^2\).

Step 3: Compute the exponent

Calculate \(4^2\). Since \(4^2 = 16\), replace \(4^2\) with 16 in the expression.

Step 4: Multiply by the coefficient

Now, multiply 5 by 16 to get the final answer. \(5 \times 16 = 80\).

Key Concepts

Substitution Method

In algebra, the substitution method allows us to replace a variable with a given value. This method is essential for evaluating expressions and solving equations. For example, if we want to evaluate the expression \(5 x^2\) for \(x = 4\), we simply replace every instance of \(x\) with 4. This means that our expression becomes \(5 \times 4^2\). This step is crucial because it sets up the rest of our calculations. Remember, substitution helps us transfer a general expression into a more specific one by using the given values.

Exponents

Exponents represent repeated multiplication of a number by itself. In the context of our example, \(4^2\) means we multiply 4 by itself, so \(4 \times 4 = 16\). This concept is crucial in algebra and higher math as it simplifies expressions and handles large numbers more efficiently. When you see an exponent, always remember it indicates how many times to use the number in a multiplication:

• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(3^2 = 3 \times 3 = 9\)
• \(5^4 = 5 \times 5 \times 5 \times 5 = 625\)

Getting comfortable with exponents will make many aspects of algebra simpler to navigate.

Multiplication

Multiplication is one of the fundamental operations in mathematics, allowing us to combine quantities in a compact way. In our expression evaluation, after calculating the exponent (\(4^2 = 16\)), the next step is to multiply this result by the coefficient, which is 5. This final step is critical: \(5 \times 16 = 80\).
When performing multiplication:

• Always start with smaller, simpler multiplications.
• Break down larger numbers into smaller factors, if necessary.
• Double-check your result to avoid errors.

Remember, the order of operations is important: do the exponentiation before the multiplication. This ensures the correct result.