Problem 126 - Elementary and Intermediate Algebra

Question

$$ \text { a. }(9 \sqrt{x-5})^{2} $$ $$ \text { b. }(9+\sqrt{x-5})^{2} $$

Step-by-Step Solution

Answer

a. \(81x - 405\) b. \(x + 76 + 18\sqrt{x-5}\)

Step 1: Understanding the Expression

The expression given is \((9\sqrt{x-5})^2\). Our task is to find the expanded form of this expression.

Step 2: Apply the Power Rule

Recall the power rule for exponents: \((ab)^n = a^n b^n\). Thus, \((9\sqrt{x-5})^2 = 9^2 (\sqrt{x-5})^2\).

Step 3: Calculate Individual Terms

Compute the terms separately: \(9^2 = 81\) and \((\sqrt{x-5})^2 = x-5\).

Step 4: Combine Terms

Multiply the results of Step 3: \(81(x-5) = 81x - 405\).

Step 5: Understanding the Second Expression

The second expression is \((9+\sqrt{x-5})^2\). We need to expand this expression using algebraic identities.

Step 6: Apply the Binomial Theorem

Recall that \((a+b)^2 = a^2 + 2ab + b^2\). Here, let \(a=9\) and \(b=\sqrt{x-5}\).

Step 7: Expand Using the Binomial Theorem

Using the identity, expand the expression: \((9+\sqrt{x-5})^2 = 9^2 + 2(9)(\sqrt{x-5}) + (\sqrt{x-5})^2\).

Step 8: Compute Individual Terms

Calculate each term: \(9^2 = 81\), \(2(9)(\sqrt{x-5}) = 18\sqrt{x-5}\), and \((\sqrt{x-5})^2 = x-5\).

Step 9: Combine Expanded Terms

Combine the expanded terms: \(81 + 18\sqrt{x-5} + x - 5\).

Step 10: Simplify the Expression

Finally, simplify the expression: the result is \(x + 76 + 18\sqrt{x-5}\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra that helps us expand expressions that involve two terms raised to a power, particularly useful for quadratic expressions. It states that when you have an expression like

• \((a + b)^2\), it can be expanded to the form \(a^2 + 2ab + b^2\).

This theorem simplifies the expansion process and is critical for solving quadratic expressions quickly and accurately.
When tackling a binomial expression such as

• \((9 + \sqrt{x-5})^2\), you assign \(a\) and \(b\) corresponding values. Here, \(a = 9\) and \(b = \sqrt{x-5}\),

and apply the theorem:

• \(9^2 + 2(9)(\sqrt{x-5}) + (\sqrt{x-5})^2\),

which then simplifies to \(81 + 18\sqrt{x-5} + x - 5\).
Ultimately, the binomial theorem bridges the gap between a complex expression and a manageable quadratic form.

Power Rule for Exponents

The power rule for exponents is an essential principle in algebra used to simplify expressions where numbers or variables are raised to a power. According to this rule, when you have a product of terms raised to an exponent, the exponent is distributed to each factor. The formula is

• \((ab)^n = a^n b^n\).

In practical terms, if you encounter an expression like

• \((9\sqrt{x-5})^2\), apply the power rule,

resulting in

• \(9^2(\sqrt{x-5})^2\).

This breaks down the complexity and allows you to compute each term separately:
\(9^2\), which equals 81, and \((\sqrt{x-5})^2\), which simplifies to \(x-5\).
The power rule transforms a daunting expression into more familiar simpler terms, facilitating ease of calculation.

Expanding Expressions

Expanding expressions is the process where algebraic expressions are simplified or rewritten without parentheses. This allows for better interpretation and manipulation of the equation or expression. It involves the application of specific algebraic rules, like the power rule and binomial theorem, to systematically break down complex mathematical expressions.
For instance, expanding the expression

• \((9\sqrt{x-5})^2\)

involves using the power rule to address the terms within the parentheses. First, separate the components and solve:

• \(9^2 = 81\)
• \((\sqrt{x-5})^2 = x - 5\)

Then, multiply the outcomes to achieve \(81(x-5)\) or \(81x - 405\).
Similarly, in the case of

• \((9+\sqrt{x-5})^2\), we apply the binomial theorem

to unravel and calculate the expanded form: \(x + 76 + 18\sqrt{x-5}\).
Expanding expressions is an indispensable skill in algebra, simplifying equations and helping to find solutions quickly and efficiently.