Problem 36 - Algebra 1

Question

Simplify the expression. $$(\sqrt{6}+5)^{2}$$

Step-by-Step Solution

Answer

The simplified expression for \((\sqrt{6}+5)^{2}\) is \(31 + 10\sqrt{6}\)

Step 1: Applying the Square of a Binomial Formula

First, apply the formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = \sqrt{6}\) and \(b = 5\). This results in the expression: \((\sqrt{6})^2 + 2\cdot\sqrt{6}\cdot5 + 5^2\).

Step 2: Solving the Squares and Multiplication

Next, calculate the squares and the multiplication in the expression: \(6 + 2\cdot\sqrt{6}\cdot5 + 25 = 6 + 10\sqrt{6} + 25\).

Step 3: Adding the Constants Together

Simplify the expression by adding the constants \(6\) and \(25\) together: \(6 + 25 = 31\), resulting in the simplified expression: \(31 + 10\sqrt{6}\).

Key Concepts

Square of a Binomial Formula

Understanding the square of a binomial formula is crucial when simplifying algebraic expressions that involve binomials raised to the second power. This formula states that for any binomial \(a + b\), the square is \(a^2 + 2ab + b^2\).

Let's break it down: the first term \(a^2\) is the square of the first term in the binomial, the middle term \(2ab\) represents twice the product of both terms, and the last term \(b^2\) is the square of the second term. Essentially, the formula encapsulates the distributive property, where you multiply each term in the binomial by every other term.

For instance, when simplifying \( (\sqrt{6}+5)^2 \) as in our exercise, we apply this formula to expand the expression before simplifying it further. This is an efficient way to handle binomial multiplication without having to multiply the binomials term by term.

Radicals in Algebra

Radicals, often recognized as 'roots', play a significant role in algebra, particularly when simplifying expressions. A radical expression involves roots, such as the square root \( \sqrt{x} \) or the cube root \( \sqrt[3]{x} \). The square root of a number is a value that, when multiplied by itself, gives the original number.

In our exercise, we encounter the square root of six \( \sqrt{6} \). When dealing with square roots in algebra, remember that \( \sqrt{x^2} = x \) for positive values of \(x\). Thus, when squaring a square root \( (\sqrt{x})^2 \) as we see with \( (\sqrt{6})^2 \), the radical sign is eliminated, and we are left with the radicand, which is the number under the root. This simplification is vital for reducing complexity in algebraic expressions.

Binomial Multiplication

Binomial multiplication is the process of multiplying two binomials together, which often involves the use of FOIL (First, Outer, Inner, Last) as a technique. However, when a binomial is multiplied by itself, as seen in squaring a binomial, we use the 'square of a binomial formula' discussed earlier.

In our example \( (\sqrt{6}+5)^2 \), rather than multiplying \(\sqrt{6}+5\) by \(\sqrt{6}+5\) individually for each term, we simplify the process by squaring both terms of the binomial and adding twice the product of the two terms. It's essential to execute each multiplication step accurately to avoid common mistakes, like forgetting to multiply the coefficient by the square root when evaluating the product \( 2\cdot\sqrt{6}\cdot5 \). Precise binomial multiplication results in a more streamlined approach to solving algebraic expressions.