Find the average of 320 , 330 , 340 .
(319 + 329 + 339 )
Calculating the Average of Powers
To find the average of a set of numbers, we sum all the numbers and then divide by the total count of numbers. In this question, we need to find the average of three numbers: \(3^{20}\), \(3^{30}\), and \(3^{40}\).
The numbers are:
- \(3^{20}\)
- \(3^{30}\)
- \(3^{40}\)
The total count of numbers is 3.
Steps to Calculate the Average
- Write down the formula for the average.
- Sum the given numbers.
- Divide the sum by the count of numbers.
- Simplify the expression using exponent rules.
The average is given by:
$$ \text{Average} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} $$
Substituting the given numbers:
$$ \text{Average} = \frac{3^{20} + 3^{30} + 3^{40}}{3} $$
We can rewrite the denominator \(3\) as \(3^1\). To simplify the expression, we can divide each term in the numerator by \(3^1\). Recall the exponent rule: \( \frac{a^m}{a^n} = a^{m-n} \).
Let's apply this rule to each term:
- First term: \( \frac{3^{20}}{3^1} = 3^{20-1} = 3^{19} \)
- Second term: \( \frac{3^{30}}{3^1} = 3^{30-1} = 3^{29} \)
- Third term: \( \frac{3^{40}}{3^1} = 3^{40-1} = 3^{39} \)
So, the average becomes the sum of these simplified terms:
$$ \text{Average} = 3^{19} + 3^{29} + 3^{39} $$
Comparing this result with the given options, we find that it matches one of them.
The calculated average of \(3^{20}\), \(3^{30}\), and \(3^{40}\) is \(3^{19} + 3^{29} + 3^{39}\).
Revision Table: Understanding Exponents
| Concept | Description | Example |
|---|---|---|
| Base | The number being multiplied by itself. | In \(3^5\), 3 is the base. |
| Exponent (Power) | The number of times the base is multiplied by itself. | In \(3^5\), 5 is the exponent. \(3^5 = 3 \times 3 \times 3 \times 3 \times 3\) |
| Division Rule | When dividing powers with the same base, subtract the exponents. \( \frac{a^m}{a^n} = a^{m-n} \) | \( \frac{5^7}{5^3} = 5^{7-3} = 5^4 \) |
Additional Information: Average Calculations and Powers
Calculating the average is a fundamental concept in mathematics used to find a central value in a set of numbers. When dealing with numbers expressed as powers, like \(3^{20}\), \(3^{30}\), and \(3^{40}\), it's important to apply the rules of exponents correctly. These numbers are very large, so calculating their exact values before finding the average would be impractical. Simplifying the expression algebraically using exponent rules, as shown above, is the correct approach.
The expression \(3^{19} + 3^{29} + 3^{39}\) represents the simplified form of the average \( \frac{3^{20} + 3^{30} + 3^{40}}{3} \).
Important Questions from Average
The sum of five numbers is 655. The average of the first two numbers is 78 and the third number is 108. Find the average of the remaining two numbers?
The average of 11 numbers is 18. Out of these 11 numbers, the average of 9 numbers is 17. If the ratio of the remaining two numbers is 2 : 3. What is the smaller number?
In a school, the average age of students is 13 years and the average age of 14 teachers is 34 years. If the average age of teachers and students combined is 14 years, then the number of students is?
The average mark of 13 papers is 80. The average marks of the first 7 papers are 84 and that of the last 7 papers is 70. Find the marks obtained in the 7th paper.
The average age of the Indian cricket team playing in the Capetown test match is 28 years. If the average age of 10 players except the Captain is 27.8 years, then the age of the Captain is:
