Problem 15 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(4 x^{2}\)

Step-by-Step Solution

Answer

(a) 64, (b) 144

Step 1 (a) - Substitute x = 4

Substitute the value of x into the expression: \[ 4x^2 = 4(4)^2 \]

Step 2 (a) - Calculate the exponent

Find the square of 4: \[ (4)^2 = 16 \]

Step 3 (a) - Multiply by 4

Multiply 4 by the result: \[ 4 \times 16 = 64 \]

Step 4 (b) - Substitute x = 6

Substitute the value of x into the expression: \[ 4x^2 = 4(6)^2 \]

Step 5 (b) - Calculate the exponent

Find the square of 6: \[ (6)^2 = 36 \]

Step 6 (b) - Multiply by 4

Multiply 4 by the result: \[ 4 \times 36 = 144 \]

Key Concepts

Substitution

Substitution is the process of replacing a variable with a given value in an algebraic expression. It's like plugging in numbers to see what happens.
For instance, in the exercise, we are asked to evaluate the expression \(4x^{2}\).
If \(x = 4\), we substitute \(4\) for \(x\). This gives us \[4(4)^{2}\text{ for part (a).}\] Similarly, for part (b), we substitute \(x = 6\). This will give us \[4(6)^{2}.\] Substitution helps in simplifying expressions and making them numerical, which you can then compute easily.

Exponentiation

Exponentiation is the mathematical operation that raises a number to a power. It's an essential concept for evaluating algebraic expressions.
For example, \(x^{2}\) means you multiply \(x\) by itself. When we have \(4x^{2}\), we first perform exponentiation before doing any multiplication.
In our problem, when \(x = 4\), \((4)^2\) calculates as \[16.\] You are squaring the 4. Similarly, when \(x = 6\), \((6)^2\) becomes \[36.\] Exponentiation must always be done before any other operations like multiplication. This follows the order of operations, often remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Multiplication

Multiplication is one of the basic arithmetic operations that combines groups of equal sizes. In evaluating algebraic expressions, once we have handled substitution and exponentiation, multiplication usually comes next.
From the given exercise, after we perform the exponentiation step, we multiply the result by 4.
For \(x = 4\), after obtaining \((4)^{2} = 16\), we multiply 4 by 16 giving us \[4 \times 16 = 64.\] Similarly, for \(x = 6\), after \((6)^{2} = 36\), we multiply 4 by 36 yielding \[4 \times 36 = 144.\] Being comfortable with multiplication ensures that we can handle more complex expressions as the problems increase in difficulty.