Problem 52 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\((9 - \sqrt{11})^2 = 92 - 18\sqrt{11}\).

Step 1: Understand the Problem

The problem requires us to square the expression \((9-\sqrt{11})\). This means we need to multiply the expression by itself.

Step 2: Apply the Square Formula

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

Step 3: Square Each Term

Calculate \(a^2 = 9^2 = 81\) and \(b^2 = (\sqrt{11})^2 = 11\).

Step 4: Calculate the Double Product

Compute \(-2ab = -2 \times 9 \times \sqrt{11} = -18\sqrt{11}\).

Step 5: Substitute and Simplify

Substitute the values from Steps 3 and 4 into the formula: \[(9 - \sqrt{11})^2 = 81 - 18\sqrt{11} + 11\] Simplify by combining like terms: \[81 + 11 = 92\] So the expression becomes \[92 - 18\sqrt{11}\].

Key Concepts

Binomial Squaring

Binomial squaring refers to the process of squaring a binomial expression, which is an algebraic expression containing two terms. In our exercise, the expression \((9 - \sqrt{11})^2\) is a binomial because it comprises two distinct terms: 9 and \(-\sqrt{11}\). Squaring such an expression involves multiplying it by itself. With binomial squaring, there are a few essential steps to follow that ensure the correct outcome.
First, it's important to recognize that when we multiply the two identical binomials, we engage both distributive property and special algebraic formulas. This means carefully handling each pair of terms, and then combining our results. The process might seem daunting at first, but using the "square formula" simplifies the overall task, which we'll delve into next.

Square Formula

The square formula is crucial to efficiently squaring binomials. The formula states:

• \((a-b)^2 = a^2 - 2ab + b^2\)

In our exercise, \(a\) is 9 and \(b\) is \(\sqrt{11}\). Applying the formula helps ensure every component gets its correct value.

Breaking Down the Formula

The square formula shows a systematic way to tackle binomial squaring:

• \(a^2\): The square of the first term. For us, this means squaring 9 to get 81.


• \(b^2\): The square of the second term. We square \(\sqrt{11}\), resulting in 11.


• \(-2ab\): The product of twice the first and second terms, ensuring proper handling of any cross-products. Here, it calculates to \(-18\sqrt{11}\).

The formula combines these three components into a single expression, making it easier to proceed to the last step: simplification.

Simplification of Expressions

Simplifying expressions is about making them clearer and easier to work with. In our given task, we've reached this stage after substituting our calculated values into the square formula.

The Steps of Simplification

When simplifying:

• Identify like terms. These are terms that include the same variable raised to the same power. In our case, the constants (numbers with no variables) \(81\) and \(11\) are like terms.


• Add or subtract these terms as needed. For our exercise, adding 81 and 11 gives us 92.


• Finally, write the expression in a clear, concise form, which here becomes \(92 - 18\sqrt{11}\).

By systematically approaching simplification, we transform a potentially complex series of operations into manageable, clear steps. This ensures the final result is not only correct but easy for others to interpret and use.