Prove that the sum of 3/4

Question

Prove that the sum of  3/4
  and  10−−√
  is irrational. Show your work, and in 1–2 sentences, explain why the sum is an irrational number.(2 points)

GPT-4o mini · Answer

To prove that $ \frac{3}{4} + \sqrt{10} $ is irrational, we will assume for contradiction that $ \frac{3}{4} + \sqrt{10} $ is rational. A rational number can be expressed as the ratio of two integers. Let $ \frac{3}{4} + \sqrt{10} = r $, where $ r $ is rational. Rearranging gives us: $$ \sqrt{10} = r - \frac{3}{4} $$ Thus, we can express $ r - \frac{3}{4} $ as a single fraction: $$ \sqrt{10} = \frac{4r - 3}{4} $$ Since $ r $ is rational, $ 4r $ is also rational, and therefore $ 4r - 3 $ is rational. Therefore, the right side of the equation is rational (as the ratio of two rational numbers). However, this leads to the conclusion that $ \sqrt{10} $ is rational because it can be expressed as a fraction of two integers, which contradicts the known fact that $ \sqrt{10} $ is irrational (as 10 is not a perfect square). Since our assumption that $ \frac{3}{4} + \sqrt{10} $ is rational leads to a contradiction, we conclude that $ \frac{3}{4} + \sqrt{10} $ must be irrational. **Explanation:** The sum of a rational number ($ \frac{3}{4} $) and an irrational number ($ \sqrt{10} $) is always irrational; thus, $ \frac{3}{4} + \sqrt{10} $ is irrational.

x · Answer

Prove that the product of 2π and 3/4 is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1–2 sentences

GPT-4o mini · Answer

To show that the product $ 2\pi \times \frac{3}{4} $ is irrational, we first calculate the product: $$ 2\pi \times \frac{3}{4} = \frac{3 \cdot 2\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2} $$ Next, we need to prove that $ \frac{3\pi}{2} $ is irrational. We know that $ \pi $ is an irrational number; hence, it cannot be expressed as a fraction of two integers. Now, let's assume for contradiction that $ \frac{3\pi}{2} $ is rational. If $ \frac{3\pi}{2} $ is rational, then we can express $ \pi $ in terms of rational numbers: $$ \pi = \frac{2r}{3} $$ where $ r $ is some rational number. Since both $ 2r $ and $ 3 $ are rational, this implies that $ \pi $ would also be a rational number, which contradicts the known fact that $ \pi $ is irrational. Thus, our assumption is incorrect, and we conclude that $ \frac{3\pi}{2} $ is irrational. **Explanation:** The product $ 2\pi \times \frac{3}{4} = \frac{3\pi}{2} $ is irrational because it is a non-zero rational multiple of $ \pi $, an irrational number, making the product also irrational.

x · Answer

Explain why it is sometimes necessary to express square roots in their simplest radical form. Then, explain the process of rewriting a square root in its simplest radical form. 1-2 sentences

GPT-4o mini · Answer

Expressing square roots in their simplest radical form is important because it makes mathematical expressions easier to understand, manipulate, and evaluate. Simplifying a square root involves factoring the number under the radical into its prime factors, identifying perfect squares, and then extracting the square roots of those perfect squares to rewrite the expression in the form $ a\sqrt{b} $, where $ a $ is a rational number and $ b $ contains no perfect square factors other than 1.

x · Answer

What must be true in order to combine radicals using addition or subtraction? Explain why it cannot always be immediately determined whether radicals can be combined with addition or subtraction. 1 sentence

GPT-4o mini · Answer

In order to combine radicals using addition or subtraction, the radicands (the expressions under the radicals) must be the same; however, it cannot always be immediately determined whether radicals can be combined because they may appear similar but represent different values or expressions that do not simplify to the same radical form.

x · Answer

List two differences between how radicals are multiplied and how they are added or subtracted. 1 sentence

GPT-4o mini · Answer

First, when multiplying radicals, you can multiply the radicands directly (i.e., $\sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b}$), while for addition or subtraction, the radicands must be the same to combine terms (i.e., $\sqrt{a} + \sqrt{a} = 2\sqrt{a}$). Second, when multiplying, the result can encapsulate the product of different radicands, but when adding or subtracting, if the radicands differ, the expression remains separate and cannot be simplified.

x · Answer

When dividing square roots, when is it beneficial to divide the radicands before expressing the fraction in its simplest radical form? What rule allows the radicands to be divided in this way?(2 points)
1 sentence

GPT-4o mini · Answer

It is beneficial to divide the radicands before simplifying a fraction when the division results in an easier expression, and this is allowed by the property that states $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, provided that $b \neq 0$.