Problem 31 - Intermediate Algebra

Question

In the following exercises, evaluate the following expressions. When \(x=2\), (a) \(x^{6}\) (b) \(4^{x}\) (c) \(2 x^{2}+3 x-7\)

Step-by-Step Solution

Answer

(a) 64, (b) 16, (c) 7

Step 1: Evaluate expression (a)

Given the expression is \(x^{6}\). Substitute \(x = 2\): \(2^{6}\). Calculate \(2^{6}\): \(2^{6} = 64\).

Step 2: Evaluate expression (b)

Given the expression is \(4^{x}\). Substitute \(x = 2\): \(4^{2}\). Calculate \(4^{2}\): \(4^{2} = 16\).

Step 3: Evaluate expression (c)

Given the expression is \(2x^{2} + 3x - 7\). Substitute \(x = 2\): \(2(2)^{2} + 3(2) - 7\). Calculate \(2(2)^{2}\): \2(4) = 8\. Calculate \3(2)\: \3(2) = 6\. Combine the results: \8 + 6 - 7 = 7\.

Key Concepts

Substitution

Substitution is a crucial concept in algebra, especially when evaluating expressions. It's essentially about replacing variables with their given values. In our example, we're given that \(x=2\). We then substitute this value into the expressions given:

For expression (a), the substitution transforms \(x^{6}\) into \(2^{6}\).

Likewise, for expression (b), the substitution changes \(4^{x}\) into \(4^{2}\).

In expression (c), the polynomial \(2x^{2} + 3x - 7\) becomes \(2(2)^{2} + 3(2) - 7\) after substitution. This step helps simplify the expressions by turning algebraic expressions into numerical calculations. Always remember, correctly substituting variables is key to accurate results!

Exponents

Exponents represent repeated multiplication of a number by itself. They are critical in various mathematical computations.

For example, \(2^{6}\) means multiplying 2 by itself six times: \[ 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \]

Similarly, \(4^{2}\) signifies multiplying 4 by itself twice: \[ 4 \times 4 = 16 \]

Understanding exponents simplifies complex expressions and makes calculations more manageable. Always ensure you correctly interpret and calculate exponents for accurate results.

Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function for a given variable. Polynomials are expressions that can include constants, variables, and exponents, combined using addition, subtraction, and multiplication.

Take the polynomial \(2x^{2} + 3x - 7\). To evaluate it at \(x=2\), we substitute 2 for every instance of x: \[ 2(2)^{2} + 3(2) - 7 = 2(4) + 6 - 7 = 8 + 6 - 7 \]
Simplifying this, we get: \[ 8 + 6 - 7 = 7 \]

This method also applies to more complex polynomials. Substituting the variable with its given value and carefully following order of operations – parentheses, exponents, multiplication, addition, and subtraction (PEMDAS) – ensures accurate results.