Problem 19 - Algebra A Combined Function

Question

Evaluate each expression with the given replacement values. $$ \frac{2 z^{4}}{5} \text { when } z=-2 $$

Step-by-Step Solution

Answer

The expression evaluates to 6.4 when \( z = -2 \).

Step 1: Understand the Expression

The expression given is \( \frac{2z^4}{5} \). The task is to evaluate this expression by substituting \( z = -2 \).

Step 2: Substitute the Value

Replace \( z \) with \( -2 \) in the expression. This gives us \( \frac{2(-2)^4}{5} \).

Step 3: Calculate the Power of z

Calculate \((-2)^4\). Since \(-2 \times -2 = 4\), then \(4 \times -2 = -8\), and \(-8 \times -2 = 16\), we get \((-2)^4 = 16\).

Step 4: Multiply by 2

Now multiply the result \(16\) by \(2\) from the numerator: \(2 \times 16 = 32\).

Step 5: Divide by 5

Divide the product by \(5\): \(\frac{32}{5} = 6.4\).

Step 6: Final Result

The final result of the expression when \( z = -2 \) is \(6.4\).

Key Concepts

Substitution Method

The substitution method is a straightforward technique used to simplify algebraic expressions by replacing a variable with its given value. In the context of our exercise, we're given the expression \( \frac{2z^4}{5} \) and told to evaluate it when \( z = -2 \). This is done by performing a direct replacement of the variable \( z \) with the value \(-2\) in the expression.

To apply substitution:

• Identify the variable in your algebraic expression.
• Replace this variable everywhere it appears in the expression with the given value.

After substitution, you'll carry out the arithmetic operations that the expression requires. In our example, it involves working with powers and division. By mastering the substitution method, you're equipped to handle a wide range of problems involving variables and can transform them into simpler numerical problems that are easier to solve.

Powers and Exponents

Powers and exponents are mathematical tools used to express repeated multiplication of the same number. When you see a number like \(-2^4\), it means \(-2\) multiplied by itself four times, which we can express as \(-2 \times -2 \times -2 \times -2\). Exponents have certain rules:

• An exponent of 2 is called a "square," and an exponent of 3 is a "cube."
• When the base is negative and the exponent is even, the result is positive.
• When the base is negative and the exponent is odd, the result is negative.

Applying this to \((-2)^4\), we calculate each step:
1. First, \(-2 \times -2 = 4\)
2. Next, \(4 \times -2 = -8\)
3. Finally, \(-8 \times -2 = 16\)
Therefore, \((-2)^4 = 16\). Recognizing these patterns helps you swiftly evaluate expressions with exponents in any problem.

Rational Expressions

Rational expressions in algebra are fractions where both the numerator and the denominator are polynomials. In our problem, \( \frac{2z^4}{5} \) is a simple rational expression. Evaluating rational expressions involves simplifying the expression using arithmetic with the given values.Let's break this down with our exercise:
1. Substitute \( z = -2 \) into \( 2z^4 \) to get \( 2(-2)^4 \).
2. Calculate the result, which is \( 2 \times 16 = 32 \), as we've already determined from the exponentiation.
3. Divide \( 32 \) by \( 5 \) to simplify the result to \( 6.4 \).Understand and practice:

• Always perform operations like powers before multiplication and division.
• In complex rational expressions, simplify both the numerator and the denominator separately if needed.

Rational expressions are common in algebra, and mastering them involves practice and careful calculation to avoid mistakes. Always perform your operations step by step to ensure accuracy.