Solved on Feb 09, 2024

Determine if each rational number $\frac{a}{b}$ is a perfect square.

STEP 1

Assumptions
1. A rational number is a number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ and $q$ are integers and $q \neq 0$ .
2. A perfect square is a number that can be expressed as the square of an integer.
3. We need to determine if each given rational number is a perfect square.

STEP 2

To determine if a rational number $\frac{p}{q}$ is a perfect square, both the numerator $p$ and the denominator $q$ must be perfect squares themselves.

STEP 3

Evaluate whether $\frac{7}{12}$ is a perfect square.

STEP 4

Check if the numerator and the denominator are perfect squares.
7 is not a perfect square because there is no integer $n$ such that $n^2 = 7$ .
12 is not a perfect square because there is no integer $n$ such that $n^2 = 12$ .

STEP 5

Since neither the numerator nor the denominator are perfect squares, $\frac{7}{12}$ is not a perfect square.

STEP 6

Evaluate whether $\frac{100}{49}$ is a perfect square.

STEP 7

Check if the numerator and the denominator are perfect squares.
100 is a perfect square because $10^2 = 100$ .
49 is a perfect square because $7^2 = 49$ .

STEP 8

Since both the numerator and the denominator are perfect squares, $\frac{100}{49}$ is a perfect square.

STEP 9

Evaluate whether 0.1 is a perfect square.

STEP 10

Convert 0.1 to a fraction.
0.1 is equivalent to $\frac{1}{10}$ .

STEP 11

Check if the numerator and the denominator are perfect squares.
1 is a perfect square because $1^2 = 1$ .
10 is not a perfect square because there is no integer $n$ such that $n^2 = 10$ .

STEP 12

Since the denominator is not a perfect square, 0.1 is not a perfect square.

STEP 13

Evaluate whether 0.01 is a perfect square.

STEP 14

Convert 0.01 to a fraction.
0.01 is equivalent to $\frac{1}{100}$ .

STEP 15

Check if the numerator and the denominator are perfect squares.
1 is a perfect square because $1^2 = 1$ .
100 is a perfect square because $10^2 = 100$ .

STEP 16

Since both the numerator and the denominator are perfect squares, 0.01 is a perfect square.
Solution:
a) $\frac{7}{12}$ is not a perfect square. b) $\frac{100}{49}$ is a perfect square. c) 0.1 is not a perfect square. d) 0.01 is a perfect square.

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Determine if each rational number $\frac{a}{b}$ is a perfect square.

STEP 1

STEP 2

To determine if a rational number $\frac{p}{q}$ is a perfect square, both the numerator $p$ and the denominator $q$ must be perfect squares themselves.

STEP 3

Evaluate whether $\frac{7}{12}$ is a perfect square.

STEP 4

Check if the numerator and the denominator are perfect squares.
7 is not a perfect square because there is no integer $n$ such that $n^2 = 7$ .
12 is not a perfect square because there is no integer $n$ such that $n^2 = 12$ .

STEP 5

Since neither the numerator nor the denominator are perfect squares, $\frac{7}{12}$ is not a perfect square.

STEP 6

Evaluate whether $\frac{100}{49}$ is a perfect square.

STEP 7

Check if the numerator and the denominator are perfect squares.
100 is a perfect square because $10^2 = 100$ .
49 is a perfect square because $7^2 = 49$ .

STEP 8

Since both the numerator and the denominator are perfect squares, $\frac{100}{49}$ is a perfect square.

STEP 9

Evaluate whether 0.1 is a perfect square.

STEP 10

Convert 0.1 to a fraction.
0.1 is equivalent to $\frac{1}{10}$ .

STEP 11

Check if the numerator and the denominator are perfect squares.
1 is a perfect square because $1^2 = 1$ .
10 is not a perfect square because there is no integer $n$ such that $n^2 = 10$ .

STEP 12

Since the denominator is not a perfect square, 0.1 is not a perfect square.

STEP 13

Evaluate whether 0.01 is a perfect square.

STEP 14

Convert 0.01 to a fraction.
0.01 is equivalent to $\frac{1}{100}$ .

STEP 15

Check if the numerator and the denominator are perfect squares.
1 is a perfect square because $1^2 = 1$ .
100 is a perfect square because $10^2 = 100$ .

STEP 16

Since both the numerator and the denominator are perfect squares, 0.01 is a perfect square.
Solution:
a) $\frac{7}{12}$ is not a perfect square. b) $\frac{100}{49}$ is a perfect square. c) 0.1 is not a perfect square. d) 0.01 is a perfect square.

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Determine if each rational number ab\frac{a}{b}ba​ is a perfect square.

STEP 1

STEP 2

To determine if a rational number pq\frac{p}{q}qp​ is a perfect square, both the numerator ppp and the denominator qqq must be perfect squares themselves.

STEP 3

Evaluate whether 712\frac{7}{12}127​ is a perfect square.

STEP 4

Check if the numerator and the denominator are perfect squares.7 is not a perfect square because there is no integer nnn such that n2=7n^2 = 7n2=7.12 is not a perfect square because there is no integer nnn such that n2=12n^2 = 12n2=12.

STEP 5

Since neither the numerator nor the denominator are perfect squares, 712\frac{7}{12}127​ is not a perfect square.

STEP 6

Evaluate whether 10049\frac{100}{49}49100​ is a perfect square.

STEP 7

Check if the numerator and the denominator are perfect squares.100 is a perfect square because 102=10010^2 = 100102=100.49 is a perfect square because 72=497^2 = 4972=49.

STEP 8

Since both the numerator and the denominator are perfect squares, 10049\frac{100}{49}49100​ is a perfect square.

STEP 9

Evaluate whether 0.1 is a perfect square.

STEP 10

Convert 0.1 to a fraction.0.1 is equivalent to 110\frac{1}{10}101​.

STEP 11

Check if the numerator and the denominator are perfect squares.1 is a perfect square because 12=11^2 = 112=1.10 is not a perfect square because there is no integer nnn such that n2=10n^2 = 10n2=10.

STEP 12

Since the denominator is not a perfect square, 0.1 is not a perfect square.

STEP 13

Evaluate whether 0.01 is a perfect square.

STEP 14

Convert 0.01 to a fraction.0.01 is equivalent to 1100\frac{1}{100}1001​.

STEP 15

Check if the numerator and the denominator are perfect squares.1 is a perfect square because 12=11^2 = 112=1.100 is a perfect square because 102=10010^2 = 100102=100.

STEP 16

Since both the numerator and the denominator are perfect squares, 0.01 is a perfect square.Solution:a) 712\frac{7}{12}127​ is not a perfect square. b) 10049\frac{100}{49}49100​ is a perfect square. c) 0.1 is not a perfect square. d) 0.01 is a perfect square.

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Determine if each rational number $\frac{a}{b}$ is a perfect square.

To determine if a rational number $\frac{p}{q}$ is a perfect square, both the numerator $p$ and the denominator $q$ must be perfect squares themselves.

Evaluate whether $\frac{7}{12}$ is a perfect square.

Check if the numerator and the denominator are perfect squares.
7 is not a perfect square because there is no integer $n$ such that $n^2 = 7$ .
12 is not a perfect square because there is no integer $n$ such that $n^2 = 12$ .

Since neither the numerator nor the denominator are perfect squares, $\frac{7}{12}$ is not a perfect square.

Evaluate whether $\frac{100}{49}$ is a perfect square.

Check if the numerator and the denominator are perfect squares.
100 is a perfect square because $10^2 = 100$ .
49 is a perfect square because $7^2 = 49$ .

Since both the numerator and the denominator are perfect squares, $\frac{100}{49}$ is a perfect square.

Convert 0.1 to a fraction.
0.1 is equivalent to $\frac{1}{10}$ .

Check if the numerator and the denominator are perfect squares.
1 is a perfect square because $1^2 = 1$ .
10 is not a perfect square because there is no integer $n$ such that $n^2 = 10$ .

Convert 0.01 to a fraction.
0.01 is equivalent to $\frac{1}{100}$ .

Check if the numerator and the denominator are perfect squares.
1 is a perfect square because $1^2 = 1$ .
100 is a perfect square because $10^2 = 100$ .

Since both the numerator and the denominator are perfect squares, 0.01 is a perfect square.
Solution:
a) $\frac{7}{12}$ is not a perfect square. b) $\frac{100}{49}$ is a perfect square. c) 0.1 is not a perfect square. d) 0.01 is a perfect square.