Question

Using the numbers 5, 8, and 24, create a problem using no more than four operations (adding, subtracting, multiplication, division, square, square root, cube, cube root) where the solution will be an irrational number. Explain why the result of your operations is an irrational number

Answer 1

The sum of rational and irrational numbers is irrational number. If we add, subtract, divide the rational and irrational numbers the result will be an irrational number.

Further explanation:

Given:

The numbers are $5$, $8$ and $24.$

Explanation:

Rational numbers are those numbers than can be written as a fraction. The decimal expansion of the rational number is terminating.

${\text{Fraction}} = \dfrac{p}{q}$

Here, $p$ and $q$ are the integers. $p$ is numerator of the fraction and $q$ is denominator of the fraction.

Irrational numbers are those numbers that cannot be written in fraction. The decimal expansion of the irrational numbers is non-terminating or recurring.

The sum of rational and irrational numbers is irrational number. If we add, subtract, divide the rational and irrational numbers the result will be an irrational number.

1. $8 \times \sqrt5 + 24$

Here, 8 is a rational number, $\sqrt 5$ is an irrational number and 24 is a rational number.

The result is the irrational number.

2. $\sqrt {\dfrac{8}{5}}- \sqrt {24}$

The result will be an irrational number as $\sqrt {\dfrac{8}{5}}$ is an irrational number. So the sum of $\sqrt {\dfrac{8}{5}}$ and $\sqrt {24}$ will be irrational.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Number System

Keywords: numbers, 5, 8, 24, irrational numbers, rational numbers, adding, subtracting, multiplication, division, square, square root, cube, cube root, operations, solution, base ten system, place value, decimal expansion, natural numbers, real numbers.

Answer 2

The given three numbers are 5,8,24.

We have to use four operations (Adding, Subtracting,multiplication, division, square, square root, cube, cube root) out of these operation to make the result an irrational number.

$1. \sqrt{\frac{24}{8}} + 5=\sqrt{3}+5,\\\\2. \sqrt{\frac{24}{8}} - 5=\sqrt{3} -5 \\\\ 3.\sqrt{\frac{24 \times 5}{8}}=\sqrt{15} \\\\ 4. (5 \times 8 \times 24)^\frac{1}{3}$

There are many more examples that you can write.

Keep in mind

1. Sum of Rational and irrational is always irrational.

2. Difference of rational and irrational is always irrational.

As none of the 5,8,24 is a perfect square , nor addition of any two results in a perfect square, so square root of their addition or subtraction  or multiplication results in an  irrational number.

Similarly ,apart from 8, none of 5,24 is a perfect cube.So if you add or subtract any number from 8 and take their cube root  or square root ,results in an irrational number.