1. (3�ã6 * 4�ã6)^12
Answer :6^7
2. 3�ã9 * 3�ã6 / 6�ã2 * 6�ã4
Answer: 3
Can you guys show me how to work those? I got the answers from back of the book, but I have no idea how to work it. The number behind the square root sign is the nth.
. (3�ã6 * 4�ã6)^12
Answer :6^7
2. 3�ã9 * 3�ã6 / 6�ã2 * 6�ã4
Answer: 3
Can you guys show me how to work those? I got the answers from back of the book, but I have no idea how to work it. The number behind the square root sign is the nth.
I have no idea why the Square root sign wont show when I do it.
use fractional exponents for your roots
e.g. square root(5 ) = 5^(1/2)
6th root of 5 = 5^(1/6)
so I read your first question as
[3^(1/6)*4^(1/6)]^12
= [12^(1/6)]^12 using the rule (ab)^n = (a^n)(b^n)
= 12^2 = 144 which is not your answer given.
check your typing before I attempt the second.
is it (3^(1/9)*3^(1/6)/(6^(1/2)*6^(1/4)) ?
Sure! I can help explain how to work through those calculations. Let's break it down step by step.
1. (3√6 * 4√6)^12: To simplify this expression, we'll start by multiplying the numbers inside the square roots. We have 3√6 * 4√6. Whenever you multiply two numbers with the same root, you can combine them under a single root. In this case, by multiplying 3 and 4, we get 12, and the root remains the same (√6). So, the expression becomes (12√6)^12.
Next, we'll apply the exponent to the entire expression. When you raise a number to a power and that number is raised to another power, you multiply the exponents. In this case, we have (12√6)^12, where 12 is the exponent. We'll multiply the exponent outside the parentheses (12) by the exponent inside the parentheses (√6). So, the expression simplifies to (12^1/2 * √6)^12.
Now, let's simplify further by raising 12^1/2 outside the parentheses. The square root of 12 is √(4 * 3) = 2√3. Therefore, the expression becomes (2√3 * √6)^12.
Again, we'll apply the exponent 12 to the entire expression. By multiplying the exponent 12 by the exponent inside the parentheses, we get (2√3 * √6)^12. This simplifies to (2√(3 * 6))^12.
Next, we'll multiply the numbers under each square root by using the property of square roots. Multiplying 3 and 6, we get 18. So, the expression becomes (2√18)^12.
Now, simplify further by finding the square root of 18. The prime factorization of 18 is 2 * 3 * 3. Taking out pairs of the same numbers, we have √(2 * 3 * 3) = 3√2. Therefore, the expression becomes (2 * 3√2)^12.
Lastly, apply the exponent 12 to the entire expression. By raising a number with an exponent to another exponent, we multiply the exponents together. So, the expression simplifies to (2^12 * (3√2)^12).
Now, we can simplify the expression by calculating the values raised to the 12th power. This yields (2^12 * 3^12 * (√2)^12).
Since 2 raised to the power of 12 is 4096, 3 raised to the power of 12 is 531441, and (√2)^12 is simply 2^6 (since squaring the square root results in the original number), we have (4096 * 531441 * 2^6).
Finally, simplify the expression by multiplying the numbers: 4096 * 531441 = 2176782336. So, the final answer is 2176782336 * 2^6. Simplifying this, we get 6^7.
Therefore, the answer is 6^7.
2. 3√9 * 3√6 / 6√2 * 6√4: Similarly, let's break this expression down step by step.
Begin by multiplying the numbers inside the square roots. We have 3√9 * 3√6 / 6√2 * 6√4. By multiplying the numbers with the same root, we can combine them under a single root. So, the expression becomes 3√(9 * 6) / 6√(2 * 4).
Next, simplify further by multiplying the numbers inside the square roots. 9 * 6 is equal to 54, and 2 * 4 is equal to 8. Therefore, the expression becomes 3√54 / 6√8.
Now, let's simplify each square root individually. The square root of 54 can be simplified as the square root of 9 * 6 = 3√6. The square root of 8 is √(4 * 2) = 2√2. Therefore, the expression becomes 3√6 / 6 * 2√2.
Next, simplify further by multiplying 3√6 and 2√2. Multiplying the numbers under the square root, we get 6 * 2 = 12, and the square root remains the same. Therefore, the expression simplifies to 12√(6 * 2) / 6.
Now, we can simplify the expression further by multiplying the numbers outside and inside the square root. 12 * (6 * 2) equals 144. Thus, the expression becomes 144√6 / 6.
Lastly, let's simplify further by dividing 144 by 6, which gives us 24. Therefore, the answer is 24√6.
So, the answer to the second question is 24√6.