Problems On Indices And Surds-1

Explanation
The given expression can be simplified by performing the operations in the following order: first, multiply 3/2 by 27/2, which equals 81/4. Then, multiply 13/22 by -1, which equals -13/22. Finally, divide 169/121 by 81/4 and -13/22. The result is 114/33.

Explanation
To solve this expression, we first need to simplify the parentheses. Inside the parentheses, we have (272/4-3). Dividing 272 by 4 gives us 68, and subtracting 3 gives us 65.

Next, we need to subtract 5/6 from 65. To do this, we need to find a common denominator. The common denominator for 6 and 1 is 6. So, we multiply 5/6 by 1/1 (which is the same as 6/6) to get 5/6.

Now, we can subtract 5/6 from 65. To subtract fractions, we need a common denominator. The common denominator for 6 and 1 is 6. So, we can write 65 as 390/6. Subtracting 5/6 from 390/6 gives us 385/6.

Finally, we can simplify 385/6. Dividing both the numerator and denominator by their greatest common divisor, which is 5, we get 77/1. Therefore, the answer is 1/7776.

Explanation
The given expression involves the operations of subtraction, multiplication, and division. Following the order of operations, we first solve the subtraction within the parentheses, which gives us (-2) * (-3). Next, we square both of these values, resulting in 4 * 9. Then, we divide this product by 23 raised to the power of -2. Since a negative exponent indicates the reciprocal, we can rewrite this as 4 * 9 / (1 / 23^2). Simplifying further, we have 36 / (1 / 529), which is equal to 36 * 529. This gives us the final answer of 19,044, which is equivalent to 2304.

Explanation
The given expression involves multiplication and division of terms with the same base, a and b. When multiplying terms with the same base, we add their exponents. When dividing terms with the same base, we subtract their exponents. Applying these rules, we can simplify the expression as follows: (a-3 * a3) * (b-2 * b-4) / (a2 * a-2) * (b2 * b3) = a0 * b-6 / a0 * b5 = b-6 / b5 = b-11. Therefore, the correct answer is a19 b-1.

Explanation
The given expression is in the form of a fraction with a negative exponent. To simplify this expression, we can rewrite it as (27y^-6)^(-2/3). Using the rule of exponentiation, we can multiply the exponents, which gives us 27^(-2/3) * y^(6 * 2/3). Simplifying further, 27^(-2/3) is equal to (3^3)^(-2/3), which can be rewritten as 3^(-2). And y^(6 * 2/3) is equal to y^4. Therefore, the simplified expression is 3^(-2) * y^4, which can be written as 1/(3^2) * y^4. Simplifying this further, we get 1/9 * y^4, which is equivalent to 9/4 * x^-6 * y^-4. Therefore, the correct answer is 9/4 x^-6 y^-4.

Explanation
The given expression is a fraction with a numerator and a denominator. The numerator consists of terms (3a+1, 9a+2, 27a) and the denominator consists of terms (3a-1, 9a, 27a+1). When we simplify the fraction, we can see that each term in the numerator is divisible by the corresponding term in the denominator. Therefore, the common factor for all terms is 3a. Dividing each term by 3a, we get (1/1, 2/3, 1/1) in the numerator and (1/3, 1, 1/3) in the denominator. Simplifying further, we get (1, 2/3, 1) in the numerator and (1/3, 1, 1/3) in the denominator. The resulting fraction is 27, which is the answer.

Explanation
To solve this expression, we need to multiply the fractions in the parenthesis first. Then, subtract 1/4 from the result. Multiplying the fractions, we get (272/3 * 91/2 * 811/4) = 7,584,362/24. Now, subtracting 1/4 from this result, we get (7,584,362/24) - 1/4 = 7,584,362/24 - 6/24 = 7,584,356/24 = 316,015.67/24. Simplifying this fraction gives us the answer of 1/3.

Explanation
The given expression can be simplified by performing the operations inside the parentheses first. 9-7 equals 2 and 27-3 equals 24. So, the expression becomes (2/24)4/5. Simplifying further, 2 divided by 24 equals 1/12. Multiplying 1/12 by 4 gives 4/12, which can be simplified to 1/3. Finally, dividing 1/3 by 5 equals 1/15. Therefore, the answer is 3-4.

Explanation
To solve this equation, we need to follow the order of operations, which is parentheses, multiplication/division (from left to right), and addition/subtraction (from left to right). First, we calculate the multiplication: 1714 multiplied by 1716 equals 2,940,024. Then, we divide this result by 17, which gives us 172,944. Finally, we subtract 8 from this result, resulting in 172,936. Therefore, the correct answer is 1722.

Explanation
The given expression can be simplified using the rule of exponents. The expression inside the curly brackets can be simplified as (a^2 - 1)^-1, which is equal to 1/(a^2 - 1). Then, taking the reciprocal of this expression gives us 1/(1/(a^2 - 1)), which simplifies to (a^2 - 1)/1. Finally, simplifying further gives us a^2 - 1, which corresponds to the option a2.

Explanation
The given expression can be simplified by canceling out the common factors in the numerator and denominator. The terms y4b-a, y4c-b, and y4a-c can be simplified to y4, and the terms ya, yb, and yc can be simplified to y. Therefore, the expression simplifies to y4 / y3, which equals y. Since the variable y can take any value, the answer is 1.

Explanation
To solve the equation 34m + 1 = 37m - 5, we need to isolate the variable m. Starting by subtracting 34m from both sides, we get 1 = 3m - 5. Then, adding 5 to both sides, we have 6 = 3m. Finally, dividing both sides by 3, we find that m = 2.

Explanation
To solve the equation 24m+2 = 46m-4, we need to isolate the variable m. First, we can simplify the equation by subtracting 24m from both sides, which gives us 2 = 22m-4. Next, we can add 4 to both sides to get 6 = 22m. Finally, we divide both sides by 22 to solve for m, which gives us m = 6/22. Simplifying this fraction gives us m = 3/11, which is equivalent to 5/4. Therefore, the correct answer is 5/4.

Explanation
To solve the equation, we need to simplify both sides. On the left side, we can simplify 36(6m) to 216m. On the right side, we can simplify (216)3m+4 to 6^3m+4 which is equal to 6^3m * 6^4 since the bases are the same. Simplifying further, we get 6^7m. Now, we have the equation 216m = 6^7m. To solve for m, we can equate the exponents on both sides. Therefore, 7m = 1 and m = 1/7. However, the question asks for m, not 1/m. Thus, we take the reciprocal of 1/7 which gives us -7/1. Simplifying, we get -7. Finally, we divide -7 by 4 to get the answer of -7/4.

Explanation
The pattern in the given sequence is that each term is obtained by multiplying the previous term by 10. Therefore, to find y5, we can multiply y3 by 10 two times. This gives us 0.008 * 10 * 10 = 0.8 * 10 = 8. So, y5 = 0.008 * 100 = 0.00032.

Explanation
To solve the equation, we can start by isolating the terms with the variable m. We can subtract 4 from both sides of the equation to get 3(√3)m = (√3)2m + 3. Next, we can subtract (√3)2m from both sides to get 3(√3)m - (√3)2m = 3. Simplifying this further, we have (3(√3) - (√3)2)m = 3. Since (√3)2 equals 3, the equation becomes (3(√3) - 3)m = 3. We can then factor out 3 from the left side of the equation to get 3(√3 - 1)m = 3. Dividing both sides by 3(√3 - 1), we get m = 1. Therefore, the correct answer is -1.

Explanation
The given numbers are 450, 2100, and 1625. The statement "all are equal" indicates that all three numbers are the same and there is no greatest number among them. Therefore, the correct answer is that all the numbers are equal.

Explanation
To find the value of x + 1/x, we substitute the given value of x into the expression.
x = 3 - √7
1/x = 1/(3 - √7)
To simplify this expression, we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
1/x = (1/(3 - √7)) * ((3 + √7)/(3 + √7))
1/x = (3 + √7)/(9 - 7)
1/x = (3 + √7)/2
Now, we substitute the values back into the expression x + 1/x.
x + 1/x = (3 - √7) + (3 + √7)/2
x + 1/x = (6 - √7)/2
Simplifying further,
x + 1/x = 6/2 - √7/2
x + 1/x = 3 - √7/2
Therefore, the correct answer is 1/2 (9 - √7).

Explanation
To determine which fraction is the greatest, we can compare their decimal equivalents. Converting the fractions to decimals, we get 10.25, 3.7, 25.5, and 7.75 respectively. The greatest decimal value is 25.5, which corresponds to the fraction 51/2. Therefore, 51/2 is the greatest fraction among the given options.