Question

Find a conjugate of each expression and the product of the expression with the conjugate. $$ 7 \sqrt{2}-2 \sqrt{7} $$

Short answer

Answer: The conjugate of the expression is $$7 \sqrt{2}+2 \sqrt{7}$$. The product of the expression and its conjugate is 70.

Step-by-step solution

Identify the conjugate

To find the conjugate of $$7 \sqrt{2}-2 \sqrt{7}$$ , we need to change the sign of its imaginary part. In this case, it's just a simple change in the sign of the second term:$$ 7 \sqrt{2}+2 \sqrt{7} $$

Multiply the expression with the conjugate

Now we need to calculate the product of the original expression and its conjugate which is $$(7\sqrt{2}-2\sqrt{7})(7\sqrt{2}+2\sqrt{7})$$ . We will use the distributive property to multiply these expressions: $$ (7 \sqrt{2} - 2 \sqrt{7} )(7 \sqrt{2} + 2 \sqrt{7})= 7\sqrt{2}\times7\sqrt{2}+7\sqrt{2}\times2\sqrt{7} -2\sqrt{7}\times7\sqrt{2} -2\sqrt{7}\times2\sqrt{7} $$

Simplify the expression

Now we can simplify the product obtained in the previous step: $$ 49(2) + 14\sqrt{14} - 14\sqrt{14} - 4(7) = 98 - 28 = 70 $$ So, the product of the expression and its conjugate is 70.

Key concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps simplify expressions, especially when dealing with parentheses. When you have an expression like \(a(b + c)\), the distributive property allows you to "distribute" the multiplication over addition. This means you multiply \(a\) by each term inside the parentheses, resulting in \(ab + ac\).
The property is particularly useful when multiplying polynomials or expressions with radicals. In our example,\((7\sqrt{2} - 2\sqrt{7})(7\sqrt{2} + 2\sqrt{7})\), the distributive property ensures each part of one expression multiplies with each part of the other.
After applying the property, you get four terms:\(7\sqrt{2}\times7\sqrt{2}, 7\sqrt{2}\times2\sqrt{7}, -2\sqrt{7}\times7\sqrt{2},\) and \(-2\sqrt{7}\times2\sqrt{7}\). Breaking down the expressions like this allows for easier combination and simplification.
Understanding and applying the distributive property correctly is key to successfully simplifying complex expressions in algebra.

Radicals

Radicals represent roots of numbers, most commonly square roots. They are written with the radical symbol \(\sqrt{}\). For example, \sqrt{2}\ is the square root of 2. Simplifying expressions involving radicals involves combining like terms and simplifying roots when possible.
In the expression \(7\sqrt{2} - 2\sqrt{7}\), each term contains a radical — \(\sqrt{2}\) and \(\sqrt{7}\) — paired with coefficients, which are 7 and -2 respectively. Multiplying expressions containing radicals follows specific rules: With same-index radicals, you multiply them within the radical. For instance, \(\sqrt{2}\times\sqrt{2} = \sqrt{4} = 2\).

• Think of simplifying radicals as reducing fractions—always aim to find the simplest form.
• When radicals don’t simplify to whole numbers, they often result in irrational numbers.

In our exercise, radicals helped contain the expression in a manageable form, even though they initially seemed more complex. These components are crucial for building and solving higher-level algebra problems.

Simplification of Expressions

Simplifying an expression involves combining like terms and reducing it to its simplest form. After using the distributive property and finding the product, the next step is to simplify. This involves recognizing and combining like terms, such as \(+14\sqrt{14}\) and \(-14\sqrt{14}\), which cancel each other out.
The expression \(49(2) + 14\sqrt{14} - 14\sqrt{14} - 4(7)\) simplifies to combining the constants: \(49(2) = 98\) and \(-4(7) = -28\), eventually giving us a straightforward result of 70.

• Always check for like terms — these are portions of the expression with identical variables or radicals.
• Canceling out terms like \(+14\sqrt{14}\) and \(-14\sqrt{14}\) is a quick way to simplify.

Simplifying expressions not only makes them easier to handle, but also helps in understanding the underlying properties in algebra. Once the process is well understood, it reveals how various mathematical components interact with each other.