Powers and Roots Practice Problems - Basic Exponents

To solve this problem, we'll focus on expressing the power $2^7$ as a series of multiplications.

• Step 1: Identify the given power expression $2^7$.
• Step 2: Convert $2^7$ into a product of repeated multiplication. This involves writing 2 multiplied by itself for a total of 7 times.
• Step 3: The expanded form of $2^7$ is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$.

By comparing this expanded form with the provided choices, we see that the correct expression is:

$2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$

Therefore, the solution to the problem is the expression that matches this expanded multiplication form, which is the choice $\text{1: } 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$.

To solve this problem, we will apply the rule of exponents:

• Any number raised to the power of 1 remains unchanged. Therefore, by the identity property of exponents, $10,000^1 = 10,000$.

Given the choices:

• $10,000 \cdot 10,000$: This is $10,000^2$.
• $10,000 \cdot 1$: Simplifying this expression yields 10,000, which is equivalent to $10,000^1$.
• $10,000 + 10,000$: This results in 20,000, not equivalent to $10,000^1$.
• $10,000 - 10,000$: This results in 0, not equivalent to $10,000^1$.

Therefore, the correct choice is $10,000 \cdot 1$, which simplifies to 10,000, making it equivalent to $10,000^1$.

Thus, the expression $10,000^1$ is equivalent to:

$10,000 \cdot 1$

To solve this problem, we'll determine the square root of the number 4.

• Step 1: Recognize that the square root of a number is asking for a value that, when multiplied by itself, yields the original number. Here, we seek a number $y$ such that $y^2 = 4$.
• Step 2: Identify that $4$ is a perfect square. The numbers $2$ and $-2$ both satisfy the equation $2^2 = 4$ and $(-2)^2 = 4$.
• Step 3: We usually consider the principal square root, which is the non-negative version. Thus, $\sqrt{4} = 2$.

Therefore, the solution to the problem is 2, which corresponds to the correct choice from the given options.

To solve this problem, we want to find the square root of 9.

Step 1: Recognize that a square root is a number which, when multiplied by itself, equals the original number. Thus, we are seeking a number $x$ such that $x^2 = 9$.

Step 2: Note that 9 is a common perfect square: $9 = 3 \times 3$. Therefore, the square root of 9 is the number that, when multiplied by itself, gives 9. This number is 3.

Step 3: Since we are interested in the principal square root, we consider only the non-negative value. Hence, the principal square root of 9 is 3.

Therefore, the solution to the problem is $3$.

To determine the square root of 16, follow these steps:

• Identify that we are looking for the square root of 16, which is a number that, when multiplied by itself, equals 16.
• Recall the basic property of perfect squares: $4 \times 4 = 16$.
• Thus, the square root of 16 is 4.

Hence, the solution to the problem is the principal square root, which is $4$.