Question

Calculate \(\left[g^{2}(\pi)-g(\pi)\right]^{1 / 3}\) if \(g(v)=|11-7 v|\).

Short answer

The value is approximately 4.784.

Step-by-step solution

Substitute the Value in the Function

First, substitute \( v = \pi \) into the function \( g(v) = |11 - 7v| \). This gives: \[ g(\pi) = |11 - 7\pi|.\]

Calculate the Value of g(π)

Now, calculate the expression inside the absolute value: \[ 11 - 7\pi. \]Using the approximation \( \pi \approx 3.14159 \): \[ 11 - 7 \times 3.14159 \approx 11 - 21.99113 = -10.99113. \]Therefore, \[ g(\pi) = |-10.99113| = 10.99113. \]

Calculate g²(π) and the Expression

Next, calculate \( g(\pi) \) squared:\[ g(\pi)^2 = (10.99113)^2 = 120.8056. \]Then find \( g^2(\pi) - g(\pi) \): \[ 120.8056 - 10.99113 = 109.81447. \]

Take the Cube Root

Finally, take the cube root of the expression:\[ (109.81447)^{1/3}. \]By calculating or using a calculator, you get \[ (109.81447)^{1/3} \approx 4.784. \]

Key concepts

Understanding the Absolute Value Function

The absolute value function, represented as \(|x|\), is a way of expressing the distance of a number from zero on a number line. Diagrammatically, this is always a non-negative value.
For instance, both \(|-3|\) and \(|3|\) equal 3 because they represent the distance of 3 units away from zero.
When dealing with expressions like \(|11 - 7v|\), it’s crucial to understand that regardless of whether \(11 - 7v\) is positive or negative, the outcome of the absolute value will always be positive.
This transforms potentially complex expressions into simpler ones by removing any negative sign.

• Always check within the absolute value first.
• Apply it to get the resulting positive number.

Decoding Cube Root Calculation

Calculating the cube root of a number involves finding a value that, when multiplied by itself three times, gives the original number.
Symbolically, the cube root of a number \(x\) is expressed as \(x^{1/3}\) or \[\sqrt[3]{x}\].
If you encounter \(109.81447^{1/3}\), you're looking for a number that, cubed, equals roughly 109.81447.
This process can often be simplified using a calculator, but understanding the concept is vital.

• Think of finding the third root as breaking down the number into smaller identical parts.
• Similar to square roots, but with an extra dimension.

Cube roots help in solving equations where variables are raised to the third power. They are critical for unraveling more intricate mathematical problems.

Mastering Substituting Values

Substituting values is the process of replacing variables with actual numbers in an expression or equation, which is essential for evaluating outcomes.
For example, in our exercise, we substitute \(v = \pi\) into the function \(g(v) = |11 - 7v|\).
This substitution enables us to calculate specific results, making the abstract more concrete.

• Firstly, ensure you know the value you are substituting. For example, \(\pi\) is often approximated as 3.14159.
• Replace the variable throughout the expression consistently.
• Calculate accordingly to obtain a specific outcome.

Substitution is a foundational skill in algebra and calculus, allowing you to turn general formulas into particular instances that can be solved numerically.

Algebraic Manipulation Insights

Algebraic manipulation is the technique of rearranging and simplifying expressions and equations to make them more solvable.
In calculus problems, this often involves combining like terms or simplifying complex expressions before executing other mathematical operations, like taking roots.
In the exercise provided, before taking the cube root, the expression \[g^2(\pi) - g(\pi)\] was calculated:
\((10.99113)^2 - 10.99113\).

• It's vital to follow each operation in the correct order: square first, then subtract.
• Maintaining a precise numeric workflow helps prevent errors, especially in multi-step problems.

This method is key to tackling problems step by step, allowing for more accurate and understandable solutions.