Question

questions 1-5 are to be solved without the use of a calculator.

1. which of the following numbers is an irrational number?

a. 6.363663666...
b. √49
c. 8.\overline{45}
d. 7\frac{5}{8}

1. how can (5²)³ be written using a single exponent?

a. 5⁻¹
b. 5⁵
c. 5⁰
d. 5⁶

1. how can 8.34 be written in the form \frac{a}{b}, where a and b are integers?

a. \frac{8.34}{1}
b. \frac{83.4}{10}
c. \frac{834}{100}
d. \frac{834}{1000}

Explanation:

Question 1

Step1: Recall irrational number definition

Irrational numbers are non - repeating, non - terminating decimals. Rational numbers can be expressed as fractions, terminating decimals, or repeating decimals.

Step2: Analyze Option A

The number \(6.363663666\cdots\) is a non - repeating, non - terminating decimal, so it is irrational.

Step3: Analyze Option B

\(\sqrt{49}=7\), which is an integer and thus a rational number (since \(7=\frac{7}{1}\)).

Step4: Analyze Option C

\(8.\overline{45}\) is a repeating decimal (the \(45\) repeats), so it is a rational number.

Step5: Analyze Option D

\(7\frac{5}{8}=\frac{7\times8 + 5}{8}=\frac{61}{8}\), which is a fraction and thus a rational number.

Step1: Recall exponent rule \((a^{m})^{n}=a^{m\times n}\)

For the expression \((5^{2})^{3}\), we use the power - of - a - power rule.

Step2: Apply the rule

Here, \(a = 5\), \(m = 2\), and \(n=3\). So \((5^{2})^{3}=5^{2\times3}=5^{6}\).

Step1: Recall how to convert a decimal to a fraction

A decimal \(x = a.bc\) (where \(a\) is the whole number part and \(bc\) is the decimal part) can be written as \(\frac{abc}{100}\) if there are two decimal places. For \(8.34\), we can think of it as \(8+\frac{34}{100}=\frac{8\times100+34}{100}=\frac{834}{100}\).

Step2: Analyze Option A

\(\frac{8.34}{1}\) is just \(8.34\) written as a fraction, but we need to express it as a fraction of integers without a decimal in the numerator.

Step3: Analyze Option B

\(\frac{83.4}{10}=8.34\), but the numerator still has a decimal, so it's not in the form of a fraction of two integers (in the strict sense of \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b
eq0\) without a decimal in \(a\)).

Step4: Analyze Option C

\(\frac{834}{100}\) is the correct fraction form of \(8.34\) since \(8.34=\frac{834}{100}\) (multiply numerator and denominator by 100 to eliminate the decimal: \(8.34\times\frac{100}{100}=\frac{834}{100}\)).

Step5: Analyze Option D

\(\frac{834}{1000}=0.834
eq8.34\)

Answer:

A. \(6.363663666\cdots\)

Question 2