Answer :
Sure, let's tackle each of these questions one by one:
1) Simplify the expression:
We have the expression [tex]\(\frac{2^0 \times 2^5}{2^4}\)[/tex].
- [tex]\(2^0 = 1\)[/tex], so [tex]\(2^0 \times 2^5 = 2^5\)[/tex].
- Now the expression becomes [tex]\(\frac{2^5}{2^4}\)[/tex].
- Using the laws of exponents, [tex]\(\frac{2^5}{2^4} = 2^{5-4} = 2^1 = 2\)[/tex].
So, the answer is [tex]\(2\)[/tex].
Correct choice: (C) 2
2) Least possible product of two different numbers from the list [tex]\(-\frac{1}{3}, 0, -7, 1, 6\)[/tex]:
To find the least possible product, you should pair the two numbers that when multiplied together give the most negative product.
- Multiplying the two most negative impactful numbers, [tex]\(-7\)[/tex] and [tex]\(6\)[/tex], gives [tex]\(-7 \times 6 = -42\)[/tex].
So, the least possible product is [tex]\(-42\)[/tex].
Correct choice: (D) -42
3) Find the value of [tex]\(0.1 + 0.11 + 0.111 + \cdots + 0.1111111111\)[/tex]:
This sequence consists of terms like [tex]\(0.1, 0.11, 0.111, \ldots\)[/tex], up to 10 terms.
The value calculated for this sequence is [tex]\(1.0987654321\)[/tex].
Correct choice: (E) 1.0987654321
4) Smallest positive integer [tex]\(k\)[/tex] such that the product of the digits of [tex]\(k^3\)[/tex] equals 36:
We need to find the smallest number [tex]\(k\)[/tex] such that when [tex]\((k^3)\)[/tex] is calculated, the product of its digits equals 36.
The smallest such [tex]\(k\)[/tex] that satisfies this condition is [tex]\(7\)[/tex].
Correct choice: (D) 7
5) Sum of [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex] given [tex]\(\frac{x}{24} = \frac{20}{2 y_1} = \frac{24}{18} = \frac{4}{z}\)[/tex]:
These represent ratios, and we solve for each variable:
- [tex]\(x = 32\)[/tex]
- [tex]\(y = 10\)[/tex]
- [tex]\(z = 18\)[/tex]
Their total sum [tex]\(x + y + z = 60\)[/tex].
Correct choice: (None, as 60 is not among the answer choices.)
6) Calculate [tex]\(Y - X\)[/tex]:
- [tex]\(X = 2 + 4 + 6 + \cdots + 90\)[/tex] is the sum of even numbers from 2 to 90.
- [tex]\(Y = 4 + 6 + 8 + \cdots + 100\)[/tex] is the sum of even numbers from 4 to 100.
The difference [tex]\(Y - X\)[/tex] is calculated as 478.
Correct choice: (A) 478
I hope this detailed breakdown helps! If you have any further questions or need more clarifications, feel free to ask.
1) Simplify the expression:
We have the expression [tex]\(\frac{2^0 \times 2^5}{2^4}\)[/tex].
- [tex]\(2^0 = 1\)[/tex], so [tex]\(2^0 \times 2^5 = 2^5\)[/tex].
- Now the expression becomes [tex]\(\frac{2^5}{2^4}\)[/tex].
- Using the laws of exponents, [tex]\(\frac{2^5}{2^4} = 2^{5-4} = 2^1 = 2\)[/tex].
So, the answer is [tex]\(2\)[/tex].
Correct choice: (C) 2
2) Least possible product of two different numbers from the list [tex]\(-\frac{1}{3}, 0, -7, 1, 6\)[/tex]:
To find the least possible product, you should pair the two numbers that when multiplied together give the most negative product.
- Multiplying the two most negative impactful numbers, [tex]\(-7\)[/tex] and [tex]\(6\)[/tex], gives [tex]\(-7 \times 6 = -42\)[/tex].
So, the least possible product is [tex]\(-42\)[/tex].
Correct choice: (D) -42
3) Find the value of [tex]\(0.1 + 0.11 + 0.111 + \cdots + 0.1111111111\)[/tex]:
This sequence consists of terms like [tex]\(0.1, 0.11, 0.111, \ldots\)[/tex], up to 10 terms.
The value calculated for this sequence is [tex]\(1.0987654321\)[/tex].
Correct choice: (E) 1.0987654321
4) Smallest positive integer [tex]\(k\)[/tex] such that the product of the digits of [tex]\(k^3\)[/tex] equals 36:
We need to find the smallest number [tex]\(k\)[/tex] such that when [tex]\((k^3)\)[/tex] is calculated, the product of its digits equals 36.
The smallest such [tex]\(k\)[/tex] that satisfies this condition is [tex]\(7\)[/tex].
Correct choice: (D) 7
5) Sum of [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex] given [tex]\(\frac{x}{24} = \frac{20}{2 y_1} = \frac{24}{18} = \frac{4}{z}\)[/tex]:
These represent ratios, and we solve for each variable:
- [tex]\(x = 32\)[/tex]
- [tex]\(y = 10\)[/tex]
- [tex]\(z = 18\)[/tex]
Their total sum [tex]\(x + y + z = 60\)[/tex].
Correct choice: (None, as 60 is not among the answer choices.)
6) Calculate [tex]\(Y - X\)[/tex]:
- [tex]\(X = 2 + 4 + 6 + \cdots + 90\)[/tex] is the sum of even numbers from 2 to 90.
- [tex]\(Y = 4 + 6 + 8 + \cdots + 100\)[/tex] is the sum of even numbers from 4 to 100.
The difference [tex]\(Y - X\)[/tex] is calculated as 478.
Correct choice: (A) 478
I hope this detailed breakdown helps! If you have any further questions or need more clarifications, feel free to ask.