QuestionIs $\sqrt{42}$ a rational number? Provide your reasoning.

Studdy Solution

STEP 1

Assumptions1. A rational number is a number that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq0$ . . An irrational number is a number that cannot be expressed as a fraction of two integers.
3. $\sqrt{42}$ is the number we are examining.

STEP 2

We need to determine if $\sqrt{42}$ can be expressed as a fraction of two integers.

STEP 3

First, consider the prime factorization of42, which is $2 \times3 \times7$ .

STEP 4

A perfect square (a square of an integer) has an even number of each of its prime factors.

STEP 5

Examine the prime factors of42. We see that each prime factor (2,3, and7) appears only once, which is an odd number of times.

STEP 6

Since42 does not have an even number of each of its prime factors, it is not a perfect square.

STEP 7

Therefore, $\sqrt{42}$ cannot be expressed as a fraction of two integers, which means it is not a rational number.
So, $\sqrt{42}$ is an irrational number.

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QuestionIs $\sqrt{42}$ a rational number? Provide your reasoning.

Studdy Solution

STEP 1

Assumptions1. A rational number is a number that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq0$ . . An irrational number is a number that cannot be expressed as a fraction of two integers.
3. $\sqrt{42}$ is the number we are examining.

STEP 2

We need to determine if $\sqrt{42}$ can be expressed as a fraction of two integers.

STEP 3

First, consider the prime factorization of42, which is $2 \times3 \times7$ .

STEP 4

A perfect square (a square of an integer) has an even number of each of its prime factors.

STEP 5

Examine the prime factors of42. We see that each prime factor (2,3, and7) appears only once, which is an odd number of times.

STEP 6

Since42 does not have an even number of each of its prime factors, it is not a perfect square.

STEP 7

Therefore, $\sqrt{42}$ cannot be expressed as a fraction of two integers, which means it is not a rational number.
So, $\sqrt{42}$ is an irrational number.

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QuestionIs 42\sqrt{42}42​ a rational number? Provide your reasoning.

Studdy Solution

STEP 1

Assumptions1. A rational number is a number that can be expressed as a fraction ab\frac{a}{b}ba​ where aaa and bbb are integers and b≠0b \neq0b=0. . An irrational number is a number that cannot be expressed as a fraction of two integers.3. 42\sqrt{42}42​ is the number we are examining.

STEP 2

We need to determine if 42\sqrt{42}42​ can be expressed as a fraction of two integers.

STEP 3

First, consider the prime factorization of42, which is 2×3×72 \times3 \times72×3×7.

STEP 4

A perfect square (a square of an integer) has an even number of each of its prime factors.

STEP 5

Examine the prime factors of42. We see that each prime factor (2,3, and7) appears only once, which is an odd number of times.

STEP 6

Since42 does not have an even number of each of its prime factors, it is not a perfect square.

STEP 7

Therefore, 42\sqrt{42}42​ cannot be expressed as a fraction of two integers, which means it is not a rational number.So, 42\sqrt{42}42​ is an irrational number.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionIs $\sqrt{42}$ a rational number? Provide your reasoning.

Assumptions1. A rational number is a number that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq0$ . . An irrational number is a number that cannot be expressed as a fraction of two integers.
3. $\sqrt{42}$ is the number we are examining.

We need to determine if $\sqrt{42}$ can be expressed as a fraction of two integers.

First, consider the prime factorization of42, which is $2 \times3 \times7$ .

Therefore, $\sqrt{42}$ cannot be expressed as a fraction of two integers, which means it is not a rational number.
So, $\sqrt{42}$ is an irrational number.