Problem 52 - Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (9-\sqrt{11})^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(92 - 18\sqrt{11}\).

Step 1: Identify the Expression

We need to simplify the expression \((9-\sqrt{11})^2\). This is a binomial expression where one term is 9 and the other is \(-\sqrt{11}\).

Step 2: Apply the Binomial Squaring Formula

Use the formula for squaring a binomial: \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = 9\) and \(b = \sqrt{11}\).

Step 3: Calculate \(a^2\)

Calculate \(9^2\), which is \(81\).

Step 4: Calculate \(-2ab\)

Calculate \(-2 \times 9 \times \sqrt{11}\), which gives \(-18\sqrt{11}\).

Step 5: Calculate \(b^2\)

Calculate \((\sqrt{11})^2\), which results in \(11\).

Step 6: Combine the results

Combine all parts from Steps 3, 4, and 5. The expression becomes \(81 - 18\sqrt{11} + 11\).

Step 7: Simplify the Expression

Combine the like terms: \(81 + 11 = 92\). Therefore, the simplified result is \(92 - 18\sqrt{11}\).

Key Concepts

Squaring Binomials

Squaring a binomial is an essential concept in algebra that involves expanding expressions of the form \((a-b)^2\). To make it simpler:

• This is where you have two terms inside a parenthesis raised to the power of two.
• The formula used to expand this is \((a-b)^2 = a^2 - 2ab + b^2\).
• In this formula, \(a\) and \(b\) are the terms inside the parenthesis.

When you apply this to the example \((9 - \sqrt{11})^2\), treat \(9\) as \(a\) and \(\sqrt{11}\) as \(b\).

• First square \(a\) to get \(a^2 = 81\).
• Next, calculate \(-2ab = -18\sqrt{11}\).
• Finally, square \(b\) to get \(b^2 = 11\).

By combining \(a^2 - 2ab + b^2\), you obtain the expanded expression. This step is crucial as it helps reveal the simplified form of binomial expressions in many algebra problems.

Radicals Simplification

Radicals, especially square roots, can often complicate expressions. Simplifying them is an important skill in algebra and can make handling expressions much easier.

• With the expression \((9-\sqrt{11})^2\), \(\sqrt{11}\) is a radical.
• When squared, \((\sqrt{11})^2 = 11\). This is because the square and the square root cancel each other out.

Identifying radicals and knowing how to simplify them helps in reducing complexity. It transforms potentially troublesome expressions into ones more manageable. Without simplification, dealing with radicals in various algebraic contexts can be challenging.

Intermediate Algebra

Intermediate algebra involves combining multiple algebraic concepts to solve more complex equations and expressions.

• This often means incorporating the squaring of binomials and the simplification of radicals, among other operations.
• For instance, from \((9 - \sqrt{11})^2\), a combination of these skills is critical to correctly expand and simplify the expression.

After applying the binomial square formula and simplifying the radicals, combining like terms is a common task. In our exercise, this involves adding the constants obtained from \(a^2\) and \(b^2\) to get \(92\) and adjust the expression accordingly.

• This last step further reduces the expression to \(92 - 18\sqrt{11}\), highlighting the importance of understanding and manipulating each part of algebraic expressions.

Mastering intermediate algebra empowers students to tackle various mathematical challenges with clarity and confidence.