Using rational approximation, what is and inequality statement comparing square root of 81 and square root of 88? Enter , or =.

Question

Using rational approximation, what is and inequality statement comparing square root of 81 and square root of 88? Enter <,>, or =.

The inequality is: square root of 81 ______ square root of 88

GPT-4o mini · Answer

To compare $\sqrt{81}$ and $\sqrt{88}$, we first calculate $\sqrt{81}$: $$ \sqrt{81} = 9 $$ Now for $\sqrt{88}$, we know it's between two perfect squares, $9^2 = 81$ and $10^2 = 100$. Let's estimate $\sqrt{88}$: $$ 9^2 = 81 \quad \text{and} \quad 10^2 = 100 $$ So, $9 < \sqrt{88} < 10$. To refine our approximation, we check the square of 9.5: $$ 9.5^2 = 90.25 $$ Since $90.25 > 88$, we test 9.4: $$ 9.4^2 = 88.36 $$ Since $88.36 > 88$, we test 9.3: $$ 9.3^2 = 86.49 $$ Now we know $9.3 < \sqrt{88} < 9.4$. Thus, $\sqrt{88}$ is slightly less than $9.4$: $$ 9 < 9.3 < \sqrt{88} < 9.4 < 10 $$ From this, we can conclude: $$ \sqrt{81} < \sqrt{88} $$ Thus, the final inequality statement is: $$ \sqrt{81} < \sqrt{88} $$

idk and idc · Answer

Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar. ____

GPT-4o mini · Answer

To express $\frac{8}{11}$ as a repeating decimal, we can perform long division. When dividing $8$ by $11$: 1. $11$ does not fit into $8$, so we write $0.$ and move to decimal places. 2. Consider $80$ (adding a decimal point and a zero). $11$ goes into $80$ $7$ times, because $11 \times 7 = 77$. 3. Subtract $77$ from $80$ to get $3$. 4. Bring down another $0$, making $30$. 5. $11$ goes into $30$ $2$ times, because $11 \times 2 = 22$. 6. Subtract $22$ from $30$ to get $8$. 7. Bring down another $0$, making it $80$ again. At this point, we see that we have returned to $80$, indicating that the digits $72$ will continue to repeat. Therefore, we can express $\frac{8}{11}$ as: $$ 0.\overline{72} $$ So the final answer is: $$ 0.\overline{72} $$

idk and idc · Answer

Convert 0.\overline{72} to a rational number in simplest form. ____

GPT-4o mini · Answer

To convert the repeating decimal $0.\overline{72}$ to a rational number, we can use the following method: Let $x = 0.\overline{72}$. Now, to eliminate the repeating decimal, multiply both sides by 100 (since the repeating part "72" has 2 digits): $$ 100x = 72.\overline{72} $$ Next, we subtract the original equation ($x = 0.\overline{72}$) from this new equation: $$ 100x - x = 72.\overline{72} - 0.\overline{72} $$ This simplifies to: $$ 99x = 72 $$ Now, solve for $x$: $$ x = \frac{72}{99} $$ Next, we simplify $\frac{72}{99}$. To do this, we find the greatest common divisor (GCD) of $72$ and $99$. The prime factorization of $72$ is $2^3 \times 3^2$, and the prime factorization of $99$ is $3^2 \times 11$. The GCD is $9$ (which is $3^2$). Now we divide both the numerator and denominator by their GCD: $$ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} $$ Thus, the rational number in simplest form equivalent to $0.\overline{72}$ is: $$ \frac{8}{11} $$

idk and idc · Answer

which of the following is a rational number between 5 and 5 1/4? A)5 1/7 B)square root of 26 C)5 2/5 D)5 1/3

GPT-4o mini · Answer

To find a rational number between $5$ and $5 \frac{1}{4}$, we can convert the values into improper fractions or decimals for easier comparison. 1. Convert $5$ and $5 \frac{1}{4}$ to improper fractions: - $5 = \frac{5 \times 4}{4} = \frac{20}{4}$ - $5 \frac{1}{4} = \frac{21}{4}$ Thus, we are looking for a rational number between $\frac{20}{4}$ and $\frac{21}{4}$. Next, let's convert the options to improper fractions or decimals: **A) $5 \frac{1}{7}$**: $$ 5 \frac{1}{7} = \frac{5 \times 7 + 1}{7} = \frac{36}{7} \approx 5.14 $$ **B) $\sqrt{26}$**: $\sqrt{25} = 5$ and $\sqrt{36} = 6$, so $\sqrt{26} \approx 5.1$, which is more than $5$ and less than $5.25$. **C) $5 \frac{2}{5}$**: $$ 5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{27}{5} = 5.4 $$ **D) $5 \frac{1}{3}$**: $$ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{16}{3} \approx 5.33 $$ Now, converting $5$ and $5 \frac{1}{4}$ to decimals: - $5 = 5.00$ - $5 \frac{1}{4} = 5.25$ Now, let's compare: - $5 \frac{1}{7} \approx 5.14$ (between $5$ and $5 \frac{1}{4}$) - $\sqrt{26} \approx 5.1$ (between $5$ and $5 \frac{1}{4}$) - $5 \frac{2}{5} = 5.4$ (greater than $5 \frac{1}{4}$) - $5 \frac{1}{3} \approx 5.33$ (greater than $5 \frac{1}{4}$) Among the choices A, B, C, and D, the following are rational numbers between $5$ and $5 \frac{1}{4}$: - **A) $5 \frac{1}{7}$** - **B) $\sqrt{26}$** So the rational numbers between $5$ and $5 \frac{1}{4}$ include: - A) **5 1/7** - B) **$\sqrt{26}$** Since the question asks for a rational number, A) $5 \frac{1}{7}$ is a clear answer. Thus, the answer is **A)** $5 \frac{1}{7}$.

idk and idc · Answer

What is a rational number with a denominator of 7 that is between the square root of 7 and the square root of 8. Write your answer as an improper fraction. _____

GPT-4o mini · Answer

To find a rational number with a denominator of 7 that is between $\sqrt{7}$ and $\sqrt{8}$, we first need to determine the approximate values of these square roots.

Calculate $\sqrt{7}$:
- Since $2.6^2 = 6.76$ and $2.7^2 = 7.29$, we know: \[ \sqrt{7} \approx 2.645 \]
Calculate $\sqrt{8}$:
- Since $2.8^2 = 7.84$ and $2.9^2 = 8.41$, we know: \[ \sqrt{8} = 2\sqrt{2} \approx 2.828 \]

Thus, we have the approximations: \[ \sqrt{7} \approx 2.645 \quad \text{and} \quad \sqrt{8} \approx 2.828 \]

Next, we look for a rational number with a denominator of $7$ that falls between these two values. We can express a rational number with denominator $7$ in the form of $ \frac{n}{7} $. To find $n$:

We want: \[ 2.645 < \frac{n}{7} < 2.828 \]

Multiplying the entire inequality by $7$: \[ 7 \times 2.645 < n < 7 \times 2.828 \]

Calculating the bounds: \[ 18.515 < n < 19.796 \]

Since $n$ must be an integer, the possible integer value for $n$ is $19$.

Thus, the rational number is: \[ \frac{19}{7} \]

To check:

Calculate $\frac{19}{7} \approx 2.714$, which indeed falls between $2.645$ and $2.828$.

Therefore, the answer is: \[ \frac{19}{7} \]

idk and idc · Answer

Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole number? Are all whole numbers also rational numbers? Give examples to support your answer.

GPT-4o mini · Answer

### Relationship Between Rational Numbers and Whole Numbers **Rational Numbers:** - A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero. This means that any number in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$, is a rational number. Examples of rational numbers include: - $\frac{1}{2}$ - $-3$ - $0$ - $\frac{5}{1} = 5$ **Whole Numbers:** - Whole numbers are a subset of the rational numbers that include all non-negative integers (0, 1, 2, 3, ...). Whole numbers do not include any negative numbers or fractions/decimals. ### Answering the Questions: 1. **Are all rational numbers also whole numbers?** - No, not all rational numbers are whole numbers. - **Example:** $\frac{1}{2}$ is a rational number because it can be expressed as the ratio of the integers 1 and 2, but it is not a whole number. - Another example is $-3$, which is a rational number but not a whole number. 2. **Are all whole numbers also rational numbers?** - Yes, all whole numbers are rational numbers. - **Example:** The whole number $5$ can be expressed as $\frac{5}{1}$, which fits the definition of a rational number. Similarly, $0$ can be expressed as $\frac{0}{1}$, and $1$ can be represented as $\frac{1}{1}$. ### Summary - **Rational Numbers** include fractions, whole numbers, and negative numbers, while **whole numbers** consist only of non-negative integers. - Thus, **all whole numbers are rational numbers**, but **not all rational numbers are whole numbers**. In conclusion, the relationship can be summarized as follows: - Whole numbers are a specific subset of rational numbers, while rational numbers encompass a broader range that includes fractions and negative numbers, in addition to whole numbers.