Question

determine whether each number is rational or irrational.

	rational	irrational
( -\frac{sqrt{16}}{sqrt{25}} )	( \bigcirc )	( \bigcirc )
( sqrt{10} )	( \bigcirc )	( \bigcirc )
( 0.overline{4} )	( \bigcirc )	( \bigcirc )

Explanation:

For \( 5\frac{3}{7} \):

Step1: Convert mixed number to improper fraction

A mixed number \( a\frac{b}{c} \) can be converted to an improper fraction as \( \frac{a\times c + b}{c} \). For \( 5\frac{3}{7} \), we have \( a = 5 \), \( b = 3 \), \( c = 7 \). So, \( 5\frac{3}{7}=\frac{5\times7 + 3}{7}=\frac{35 + 3}{7}=\frac{38}{7} \).

Step2: Determine if rational

A rational number is a number that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q
eq0 \). Here, \( \frac{38}{7} \) is in the form of \( \frac{p}{q} \) with \( p = 38 \) and \( q = 7 \) (both integers, \( q
eq0 \)). So, \( 5\frac{3}{7} \) is rational.

For \( -\frac{\sqrt{16}}{\sqrt{25}} \):

Step1: Simplify square roots

We know that \( \sqrt{16}=4 \) (since \( 4\times4 = 16 \)) and \( \sqrt{25}=5 \) (since \( 5\times5 = 25 \)).

Step2: Simplify the fraction

Substitute the values of the square roots into the expression: \( -\frac{\sqrt{16}}{\sqrt{25}}=-\frac{4}{5} \).

Step3: Determine if rational

\( -\frac{4}{5} \) is in the form \( \frac{p}{q} \) with \( p=- 4 \) and \( q = 5 \) (both integers, \( q
eq0 \)). So, \( -\frac{\sqrt{16}}{\sqrt{25}} \) is rational.

For \( \sqrt{10} \):

Step1: Check if 10 is a perfect square

A perfect square is a number that is the square of an integer. The square of integers near \( \sqrt{10} \): \( 3^2=9 \) and \( 4^2 = 16 \). Since \( 9<10<16 \) and there is no integer \( n \) such that \( n^2=10 \), \( \sqrt{10} \) is a non - repeating, non - terminating decimal.

Step2: Determine if rational

A rational number can be expressed as a fraction \( \frac{p}{q} \) ( \( p,q\in\mathbb{Z},q
eq0 \) ). Since \( \sqrt{10} \) cannot be written in this form (it is an irrational number by definition of square roots of non - perfect squares), \( \sqrt{10} \) is irrational.

For \( 0.\overline{4} \):

Answer:

• \( 5\frac{3}{7} \): Rational (circle the "Rational" option for \( 5\frac{3}{7} \))
• \( -\frac{\sqrt{16}}{\sqrt{25}} \): Rational (circle the "Rational" option for \( -\frac{\sqrt{16}}{\sqrt{25}} \))
• \( \sqrt{10} \): Irrational (circle the "Irrational" option for \( \sqrt{10} \))
• \( 0.\overline{4} \): Rational (circle the "Rational" option for \( 0.\overline{4} \))