practice: squaring fractions and decimals multiplying a number by itself is called squaring. 5 squared means 5 × 5. you can use an exponent to write the same thing. $5^2 = 5 \times 5 = 25$. square these fractions or decimals: 1. $\left(\frac{1}{2}\
ight)^2 =$ 2. $.05^2 =$ 3. $\left(9\frac{1}{3}\
ight)^2 =$ 4. $(2.5)^2 =$ 5. $(.2)^2 =$ 6. $(.02)^2 =$ 7. $(5.8)^2 =$ 8. $\left(\frac{5}{6}\
ight)^2 =$ complete the following sentences: 9. when you square a number less than 1, the result is a ______ 10. when you square a number greater than 1, the result is ______ 11. what other patterns do you notice?

Question

Accepted Answer

### 1. $(\frac{1}{2})^2$ # Explanation: ## Step1: Recall squaring a fraction rule To square a fraction $\frac{a}{b}$, we square the numerator and the denominator separately, i.e., $(\frac{a}{b})^2=\frac{a^2}{b^2}$. ## Step2: Apply the rule For $(\frac{1}{2})^2$, $a = 1$, $b = 2$. So we have $\frac{1^2}{2^2}=\frac{1}{4}$. # Answer: $\frac{1}{4}$ ### 2. $.05^2$ # Explanation: ## Step1: Recall squaring a decimal rule To square a decimal, we multiply the decimal by itself. ## Step2: Perform the multiplication $.05 imes.05 = 0.0025$. # Answer: $0.0025$ ### 3. $(9\frac{1}{3})^2$ # Explanation: ## Step1: Convert mixed number to improper fraction $9\frac{1}{3}=\frac{9 imes3 + 1}{3}=\frac{28}{3}$. ## Step2: Square the improper fraction Using the rule for squaring a fraction $(\frac{a}{b})^2=\frac{a^2}{b^2}$, here $a = 28$, $b = 3$. So $(\frac{28}{3})^2=\frac{28^2}{3^2}=\frac{784}{9}=87\frac{1}{9}\approx87.111\cdots$ # Answer: $\frac{784}{9}$ (or $87\frac{1}{9}$ or approximately $87.11$) ### 4. $(2.5)^2$ # Explanation: ## Step1: Recall squaring a decimal rule Multiply the decimal by itself. ## Step2: Perform the multiplication $2.5 imes2.5 = 6.25$. # Answer: $6.25$ ### 5. $(.2)^2$ # Explanation: ## Step1: Recall squaring a decimal rule Multiply the decimal by itself. ## Step2: Perform the multiplication $.2 imes.2 = 0.04$. # Answer: $0.04$ ### 6. $(.02)^2$ # Explanation: ## Step1: Recall squaring a decimal rule Multiply the decimal by itself. ## Step2: Perform the multiplication $.02 imes.02 = 0.0004$. # Answer: $0.0004$ ### 7. $(5.8)^2$ # Explanation: ## Step1: Recall squaring a decimal rule Multiply the decimal by itself. ## Step2: Perform the multiplication $5.8 imes5.8 = 33.64$. # Answer: $33.64$ ### 8. $(\frac{5}{6})^2$ # Explanation: ## Step1: Recall squaring a fraction rule For a fraction $\frac{a}{b}$, $(\frac{a}{b})^2=\frac{a^2}{b^2}$. ## Step2: Apply the rule Here $a = 5$, $b = 6$. So $(\frac{5}{6})^2=\frac{5^2}{6^2}=\frac{25}{36}\approx0.6944\cdots$ # Answer: $\frac{25}{36}$ (or approximately $0.694$) ### 9. When you square a number less than 1, the result is a number that is smaller than the original number (since multiplying a number between 0 and 1 by itself makes it smaller, e.g., $0.5^2 = 0.25<0.5$, $0.2^2=0.04 < 0.2$). # Brief Explanations: By examining examples like squaring 0.5 (result 0.25), 0.2 (result 0.04), we see that squaring a number between 0 and 1 gives a smaller number. # Answer: a number smaller than the original number (e.g., "smaller than the number you started with", "less than the original number") ### 10. When you square a number greater than 1, the result is a number that is larger than the original number (since multiplying a number greater than 1 by itself makes it larger, e.g., $2^2 = 4>2$, $3^2=9 > 3$, $2.5^2 = 6.25>2.5$). # Brief Explanations: From examples like squaring 2 (result 4), 3 (result 9), 2.5 (result 6.25), we observe that squaring a number greater than 1 gives a larger number. # Answer: a number larger than the original number (e.g., "larger than the number you started with", "greater than the original number") ### 11. Other patterns: - Squaring a negative number (if we consider negative numbers) gives a positive result (though the given problems have non - negative numbers, in general $(-a)^2=a^2$). - Squaring a fraction $\frac{a}{b}$ (where $a,b>0$) gives a fraction $\frac{a^2}{b^2}$, and if $a < b$, the squared fraction is smaller than the original fraction (consistent with the "less than 1" case), if $a>b$, the squared fraction is larger than the original fraction (consistent with the "greater than 1" case). - For decimals, the number of decimal places in the square is twice the number of decimal places in the original decimal (e.g., $.05$ has 2 decimal places, $.05^2 = 0.0025$ has 4 decimal places; $.2$ has 1 decimal place, $.2^2=0.04$ has 2 decimal places). # Brief Explanations: We can observe from the given problems and general number properties. For example, the decimal place pattern: in $.05^2$, original decimal places = 2, squared decimal places = 4; in $.2^2$, original decimal places = 1, squared decimal places = 2. For fractions, the relationship between the numerator, denominator and the size of the squared fraction. # Answer: - The number of decimal places in the square of a decimal is twice the number of decimal places in the original decimal. - Squaring a fraction $\frac{a}{b}$ gives $\frac{a^2}{b^2}$, and the relative size to the original fraction depends on whether $a < b$ (smaller when squared) or $a>b$ (larger when squared). - Squaring a negative number (in general) gives a positive result. (Any valid pattern noticed from the problems or number theory can be written here)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1. $(\frac{1}{2})^2$

Step1: Recall squaring a fraction rule

Step2: Apply the rule

Step1: Recall squaring a decimal rule

Step2: Perform the multiplication

Step1: Convert mixed number to improper fraction

Step2: Square the improper fraction

Answer:

2. $.05^2$