Basis of a power - Examples, Exercises and Solutions

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 follow these steps:

• Step 1: Recognize that $6^2$ means $6 \times 6$.
• Step 2: Perform the multiplication of 6 by itself.

Now, let's work through each step:
Step 1: The expression $6^2$ indicates we need to multiply 6 by itself.
Step 2: Calculating $6 \times 6$ gives us 36.

Therefore, the value of $6^2$ is 36.

To solve this problem, we'll follow these steps:

• Step 1: Set up the multiplication as $11 \times 11$.
• Step 2: Compute the product using basic arithmetic.
• Step 3: Compare the result with the provided multiple-choice answers to identify the correct one.

Now, let's work through each step:
Step 1: We begin with the calculation $11 \times 11$.
Step 2: Perform the multiplication:

1. Multiply the units digits: $1 \times 1 = 1$.
2. Next, for the tens digits: $11 \times 10 = 110$.
3. Add the results: $110 + 1 = 111$. This doesn't seem right, so let's break it down further.

Let's examine a more structured multiplication method:

Multiply $11$ by $1$ (last digit of the second 11), we get 11.
Multiply $11$ by $10$ (tens place of the second 11), we get 110.

If we align correctly and add the partial products:

     11
+   110
------------
   121

Step 3: The correct multiplication yields the final result as $121$. Upon reviewing the provided choices, the correct answer is choice 4: $121$.

Therefore, the solution to the problem is $121$.

To determine which expression equals 100, we need to evaluate each option:

• Option 1: $5^2\cdot5$
  - Calculate $5^2 = 25$.
  - Then compute $25 \cdot 5 = 125$.
• Option 2: $4^2\cdot4$
  - Calculate $4^2 = 16$.
  - Then compute $16 \cdot 4 = 64$.
• Option 3: $25^4$
  - Calculate $(25^4)$, which simplified through breakdown is larger than 100 because $25^2 = 625$. Hence this 25 to the power of 4 will definitely be much larger than 100.
• Option 4: $5^2\cdot2^2$
  - Calculate $5^2 = 25$.
  - Calculate $2^2 = 4$.
  - Compute $25 \cdot 4 = 100$.

Therefore, the expression in Option 4, $5^2\cdot2^2$, equals 100. Thus, the correct choice is 4.

Thus, the clause that equals 100 is $5^2\cdot2^2$.

To solve the problem, let's represent the repeated multiplication using exponents:

We start with the given expression:

$\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$

Notice that $\frac{1}{5}$ is multiplied by itself four times. This can be expressed as a power:

$(\frac{1}{5})^4$

Hence, the correct representation of the given expression is $(\frac{1}{5})^4$.

From the given choices, the correct option is Choice 4: $(\frac{1}{5})^4$.

Therefore, the solution to the problem is $(\frac{1}{5})^4$.

We use the formula: $a\times a=a^2$

In the formula, we see that the power shows the number of terms that are multiplied, that is, two times

Since in the exercise we multiply 6 three times, it means that we have 3 terms.

Therefore, the power, which is n in this case, will be 3.

Remember that according to the order of operations, exponents come before multiplication and division, which come before addition and subtraction (and parentheses always before everything),

So first calculate the values of the terms in the power and then subtract between the results:

$3^2-3^3 =9-27=-18$Therefore, the correct answer is option A.

Remember that according to the order of operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms in the power and then subtract between the results:

$3^2+3^3 =9+27=36$Therefore, the correct answer is option B.

To solve the problem, follow these steps:

• Step 1: Begin with the equation $\sqrt{x} = 2$.
• Step 2: Square both sides of the equation to eliminate the square root.
• Step 3: Simplify the resulting equation to find $x$.

Now, let's proceed through each step:
Step 1: The given equation is $\sqrt{x} = 2$.
Step 2: Square both sides: $(\sqrt{x})^2 = 2^2$.
Step 3: This simplifies to $x = 4$.

Therefore, the value of $x$ that satisfies $\sqrt{x} = 2$ is $x = 4$.

Matching this solution with the provided choices, the correct answer is choice 3, which is 4.

To solve this problem, we will perform the following steps:

• Step 1: Square both sides of the equation $\sqrt{x} = 6$.
• Step 2: Simplify the equation to find $x$.

Let's carry out each step in detail:

Step 1: Square both sides of the equation:
$\ (\sqrt{x})^2 = 6^2$

Step 2: Simplify the equation:
Since $(\sqrt{x})^2 = x$, we have $x = 36$.

Therefore, the value of $x$ is 36.

To solve this problem, we'll evaluate $5^3$, which means we need to calculate the product of multiplying the number $5$ by itself three times.

• Step 1: Compute $5 \times 5$.
• Step 2: Compute the result of $5 \times 5$ and multiply it by $5$ again.

Let's work through each step:

Step 1: Calculate $5 \times 5 = 25$.

Step 2: Take the result $25$ and multiply it by $5$:
$25 \times 5 = 125$.

Therefore, the value of $5^3$ is $125$.

To solve the problem of finding $7^3$, follow these steps:

• Step 1: Understand that the power $7^3$ means 7 multiplied by itself three times.
• Step 2: First, compute $7 \times 7$. This equals 49.
• Step 3: Next, multiply the result by 7 again: $49 \times 7$.
• Step 4: Calculate $49 \times 7$, which equals 343.

Therefore, $7^3 = 343$.

With the possible choices given, the correct answer corresponds to choice $343$.

To solve this problem, we'll follow the steps below:

• Step 1: Start with the given equation: $\sqrt{x} = 14$.
• Step 2: Square both sides of the equation to eliminate the square root:

$(\sqrt{x})^2 = 14^2$

Step 3: Calculate the square of 14:
$14 \times 14 = 196$

Therefore, the value of $x$ is 196.

Comparing our solution with the provided choices, choice 3 ($196$) is the correct match.

Thus, the solution to the problem is $x = 196$.

To solve the given problem, we will follow these steps:

• Step 1: Square both sides of the equation
• Step 2: Simplify to find $x$

Now, let's work through each step:

Step 1: We are given $\sqrt{x} = 15$.
To eliminate the square root, square both sides of the equation:

Step 2: Simplify both sides:
On the left, $(\sqrt{x})^2 = x$.
On the right, $15^2 = 225$.

This gives us the equation:

Thus, the solution to the problem is $\boxed{225}$.