Problem 51 - Elementary and Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (6-\sqrt{3})^{2} $$

Step-by-Step Solution

Answer

The simplified result is \(39 - 12\sqrt{3}\).

Step 1: Identify the Formula Needed

To square the expression \((6-\sqrt{3})^{2}\), we will use the binomial square formula: \((a-b)^{2} = a^{2} - 2ab + b^{2}\) where \(a = 6\) and \(b = \sqrt{3}\).

Step 2: Square Each Term

First, square each of the individual terms in the expression:\[6^{2} = 36\]\[(\sqrt{3})^{2} = 3\]

Step 3: Multiply the Terms and Double the Product

Calculate the product of the terms \(a\) and \(b\), and then multiply by 2, as per the formula:\[2 \times 6 \times \sqrt{3} = 12\sqrt{3}\]

Step 4: Substitute Back into the Formula

Using the formula \((a-b)^{2} = a^{2} - 2ab + b^{2}\), substitute the values we've found:\[a^{2} = 36\] \[-2ab = -12\sqrt{3}\] \[b^{2} = 3\]Plug these values into the formula: \[(6-\sqrt{3})^{2} = 36 - 12\sqrt{3} + 3\]

Step 5: Simplify the Expression

Combine the like terms to simplify the expression:\[36 + 3 = 39\]Thus, the simplified expression is:\[39 - 12\sqrt{3}\]

Key Concepts

Binomial Expansion

Binomial Expansion is a way to expand expressions that are raised to a power, particularly those involving two terms. Think of it like unfolding a package, revealing each part one at a time. In mathematics, when you have an expression like \[(a-b)^2\] or even \[(a+b)^2\], we use a special group of formulas to break these down easily. This process saves you from doing all the math from scratch every time.
Some key points to remember are:

• For \((a+b)^2\), the formula is \(a^2 + 2ab + b^2\).
• For \((a-b)^2\), the formula changes slightly to \(a^2 - 2ab + b^2\).

This technique is powerful for calculations and is frequently seen in algebra.
It helps us simplify quickly and effectively for each different problem we face.

Simplifying Expressions

Simplifying Expressions is like cleaning your room—you're organizing things to make them look neat and easy to manage. In algebra, simplifying means combining terms or making an expression as compact as possible without changing its value.
The main goals of simplifying are:

• To combine like terms, which means putting together numbers and variables that look the same.
• To apply operations like addition, subtraction, and sometimes multiplication or division, on constants or like terms.

It's like solving a puzzle where pieces fit together perfectly to show the simplest picture. Keeping track of these terms ensures we achieve the most straightforward form.

Understanding Exponents

Exponents might seem confusing at first, but they are incredibly useful. They let you show repeated multiplication more succinctly.
Here’s a quick look at how exponents work:

• If \(x^2\) is in an expression, it means \(x imes x\).
• An exponent of 3, like \(x^3\), means \(x imes x imes x\).
• Raising to the power of 0, like \(x^0\), will always equal 1, no matter what x is.

Understanding exponents furthers your grasp of algebra because they pop up everywhere—from simplifying expressions to solving equations. Mastery of exponents allows you to manage big numbers, plus makes those tedious calculations much cleaner and quicker.