Problem 60 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (2 a+\sqrt{5 a})^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((2a+\sqrt{5a})^2\). Answer: \(4a^2 + 4a\sqrt{5a} + 5a\)

Step 1: Identify the terms A and B

We identify A and B in the given expression, where A = \(2a\) and B = \(\sqrt{5a}\)

Step 2: Apply the binomial square formula

Now, we'll apply the binomial square formula: \((A+B)^2 = A^2 + 2AB + B^2\)

Step 3: Substitute A and B values and simplify

Substituting the values of A and B, we have: \((2a+\sqrt{5a})^2 = (2a)^2 + 2(2a)(\sqrt{5a}) + (\sqrt{5a})^2\) Now, let's simplify each term: 1. \((2a)^2 = 4a^2\) 2. \(2(2a)(\sqrt{5a}) = 4a\sqrt{5a}\) 3. \((\sqrt{5a})^2 = 5a\)

Step 4: Combine the simplified terms

Now, combine the simplified terms to get the final answer: \((2a+\sqrt{5a})^2 = 4a^2 + 4a\sqrt{5a} + 5a\) The simplified expression is: $$ (2a+\sqrt{5a})^2 = 4a^2 + 4a\sqrt{5a} + 5a $$

Key Concepts

Simplifying Algebraic Expressions

When tackling algebra, simplifying expressions is a fundamental skill that paves the way towards understanding more complex mathematical concepts. Simplifying an algebraic expression means to make it as easy to read and as compact as possible, without changing its value.

In the context of our exercise \( (2a + \sqrt{5a})^2 \) , we first identify individual terms and then use algebraic rules to combine like terms and reduce complexity. For instance, squaring the term \(2a\) turns it into \(4a^2\)\), and squaring the radical expression \(\sqrt{5a}\)\) eliminates the radical, giving \(5a\)\).

Simplification often involves expanding expressions using distributive properties, combining like terms, and sometimes factoring. In this exercise, we avoided factoring but effectively used distribution to expand the binomial and then combine similar terms to achieve a simplified answer.

Radical Expressions

A radical expression involves roots, such as square roots or cube roots. In algebra, we often encounter square roots, which are indicated by the radical symbol \(\sqrt{\ }\)\).

Simplifying radical expressions can include rationalizing the denominator, combining like radicals, or, as in our step-by-step solution, squaring a radical to eliminate the square root.

Rationalize and Combine

If you have a term under a radical, it can often be simplified by multiplying and dividing by a 'conjugate' to eliminate the radical from the denominator. Also, like radicals (those with the same radicand, or number under the radical) can often be combined through addition or subtraction.

In our case, \(\sqrt{5a}\)\) is a radical expression squared in the process. Squaring the root and the number inside it—\(5a\)\)—gives us a non-radical expression of \(5a\)\), simplifying the overall expression.

Polynomial Expansion

Polynomial expansion is the process of multiplying polynomials to remove parentheses. This operation increases the degree of the polynomial or changes its terms. The binomial square formula—\((A+B)^2 = A^2 + 2AB + B^2\)—is a commonly used expansion for binomials, a polynomial with two terms.

In our exercise, expansion allows us to first square each term inside the parentheses using the formula, yielding separate simpler expressions we can combine for our final answer.

Work Through Each Term

The formula tells us to first square the first term (A), multiply the first term by the second term and double it (2AB), and finally square the second term (B). After applying the formula, we combine like terms if possible to simplify further.

Mastery of polynomial expansion helps in understanding more advanced algebra, calculus, and beyond. In this exercise, correctly applying the binomial square formula has been crucial for expanding and simplifying the given expression.