Exponents and Roots

What is an exponent?

An exponent tells us the amount of times a number is to be multiplied by itself.

What is a root?

A root is the inverse operation of exponentiation, which helps us discover which number multiplied by itself gives this result.

The square root is equal to the power of 0.5.

Exponents and the Base of the Exponents

Practice Powers and Roots - Basic

Examples with solutions for Powers and Roots - Basic

Exercise #1

112= 11^2=

Video Solution

Step-by-Step Solution

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

  • Step 1: Set up the multiplication as 11×11 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 11×11 11 \times 11 .
Step 2: Perform the multiplication:

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

Let's examine a more structured multiplication method:

Multiply 11 11 by 1 1 (last digit of the second 11), we get 11.
Multiply 11 11 by 10 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 121 121 . Upon reviewing the provided choices, the correct answer is choice 4: 121 121 .

Therefore, the solution to the problem is 121 121 .

Answer

121

Exercise #2

62= 6^2=

Video Solution

Step-by-Step Solution

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

  • Step 1: Recognize that 62 6^2 means 6×6 6 \times 6 .
  • Step 2: Perform the multiplication of 6 by itself.

Now, let's work through each step:
Step 1: The expression 62 6^2 indicates we need to multiply 6 by itself.
Step 2: Calculating 6×6 6 \times 6 gives us 36.

Therefore, the value of 62 6^2 is 36.

Answer

36

Exercise #3

64= \sqrt{64}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Recognize what finding a square root means
  • Step 2: List known perfect squares to identify which one results in 64
  • Step 3: Verify the square root by calculation

Now, let's work through each step:
Step 1: To find the square root of 64, we seek a number that, when multiplied by itself, equals 64.
Step 2: Consider the sequence of perfect squares: 12=1 1^2 = 1 , 22=4 2^2 = 4 , 32=9 3^2 = 9 , 42=16 4^2 = 16 , 52=25 5^2 = 25 , 62=36 6^2 = 36 , 72=49 7^2 = 49 , 82=64 8^2 = 64 .
Step 3: We see that 82=64 8^2 = 64 . Therefore, the square root of 64 is 8.

Therefore, the solution to this problem is 8 8 .

Answer

8

Exercise #4

36= \sqrt{36}=

Video Solution

Step-by-Step Solution

Let's solve the problem step by step:

  • Step 1: Understand what the square root means.
  • The square root of a number nn is a value that, when multiplied by itself, equals nn. This is written as x=nx = \sqrt{n}.

  • Step 2: Apply this definition to the number 3636.
  • We are looking for a number xx such that x2=36x^2 = 36. This translates to finding x=36x = \sqrt{36}.

  • Step 3: Determine the correct number.
  • We know that 6×6=366 \times 6 = 36. Therefore, the principal square root of 3636 is 66.

Thus, the solution to the problem is 36=6 \sqrt{36} = 6 .

Among the given choices, the correct one is: Choice 1: 66.

Answer

6

Exercise #5

49= \sqrt{49}=

Video Solution

Step-by-Step Solution

To solve this problem, we follow these steps:

  • Step 1: Understand that finding the square root of a number means determining what number, when multiplied by itself, equals the original number.
  • Step 2: Identify the numbers that could potentially be the square root of 4949. These are ±7 \pm7, but by convention, the square root function typically refers to the non-negative root.
  • Step 3: Calculate 7×7=497 \times 7 = 49. This confirms that 49=7 \sqrt{49} = 7.
  • Step 4: Verify using the problem's multiple-choice answers to ensure 77 is among them, confirming choice number .

Therefore, the solution to the problem 49 \sqrt{49} is 7 7 .

Answer

7

Exercise #6

Which of the following is equivalent to the expression below?

10,0001 10,000^1

Video Solution

Step-by-Step Solution

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, 10,0001=10,000 10,000^1 = 10,000 .

Given the choices:

  • 10,00010,000 10,000 \cdot 10,000 : This is 10,0002 10,000^2 .
  • 10,0001 10,000 \cdot 1 : Simplifying this expression yields 10,000, which is equivalent to 10,0001 10,000^1 .
  • 10,000+10,000 10,000 + 10,000 : This results in 20,000, not equivalent to 10,0001 10,000^1 .
  • 10,00010,000 10,000 - 10,000 : This results in 0, not equivalent to 10,0001 10,000^1 .

Therefore, the correct choice is 10,0001 10,000 \cdot 1 , which simplifies to 10,000, making it equivalent to 10,0001 10,000^1 .

Thus, the expression 10,0001 10,000^1 is equivalent to:

10,0001 10,000 \cdot 1

Answer

10,0001 10,000\cdot1

Exercise #7

Choose the expression that is equal to the following:

27 2^7

Video Solution

Step-by-Step Solution

To solve this problem, we'll focus on expressing the power 27 2^7 as a series of multiplications.

  • Step 1: Identify the given power expression 27 2^7 .
  • Step 2: Convert 27 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 27 2^7 is 2×2×2×2×2×2×2 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:

2222222 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 1: 2222222\text{1: } 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2.

Answer

2222222 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2

Exercise #8

Choose the largest value

Video Solution

Step-by-Step Solution

Let's begin by calculating the numerical value of each of the roots in the given options:

25=516=49=3 \sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\ We can determine that:

5>4>3>1 Therefore, the correct answer is option A

Answer

25 \sqrt{25}

Exercise #9

Sovle:

32+33 3^2+3^3

Video Solution

Step-by-Step Solution

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:

32+33=9+27=36 3^2+3^3 =9+27=36 Therefore, the correct answer is option B.

Answer

36

Exercise #10

121= \sqrt{121}=

Video Solution

Step-by-Step Solution

To solve the problem of finding the square root of 121, let's follow these steps:

  • Step 1: Understand that we need to find a number x x such that x2=121 x^2 = 121 .
  • Step 2: Recognize that 121 is a perfect square. Specifically, 11×11=121 11 \times 11 = 121 .
  • Step 3: Therefore, the square root of 121 is clearly 11 11 .

Thus, the solution to the problem is 11 11 .

Answer

11

Exercise #11

100= \sqrt{100}=

Video Solution

Step-by-Step Solution

The task is to find the square root of the number 100. The square root operation seeks a number which, when squared, equals the original number. For any positive integer, if x2=100 x^2 = 100 , then x x should be our answer.

Step 1: Recognize that 100 is a perfect square. This means there exists an integer x x such that x×x=100 x \times x = 100 . Generally, we recall basic squares such as:

  • 12=1 1^2 = 1
  • 22=4 2^2 = 4
  • 32=9 3^2 = 9
  • and so forth, up to 102 10^2

Step 2: Checking integers, we find that:

102=10×10=100 10^2 = 10 \times 10 = 100

Step 3: Confirm the result: Since 10×10=100 10 \times 10 = 100 , then 100=10 \sqrt{100} = 10 .

Step 4: Compare with answer choices. Given that one of the choices is 10, and 100=10 \sqrt{100} = 10 , choice 1 is correct.

Therefore, the square root of 100 is 10.

Answer

10

Exercise #12

144= \sqrt{144}=

Video Solution

Step-by-Step Solution

To solve this problem, we proceed with the following steps:

  • Step 1: Identify that the problem asks for the square root of 144.
  • Step 2: Use the concept of square roots, where n=x\sqrt{n} = x implies x2=nx^2 = n.
  • Step 3: Check for a perfect square whose square equals 144.

Now, let's solve the problem:

Step 1: We need the square root 144\sqrt{144}.

Step 2: Recall that x2=144x^2 = 144. We need to find xx.

Step 3: Recognize that 144 is a perfect square and find xx such that x2=144x^2 = 144. Through either calculation or prior knowledge, we know:
122=14412^2 = 144.

Therefore, the square root of 144 is 144=12\sqrt{144} = 12.

Thus, the solution to the problem is 12\mathbf{12}.

Answer

12

Exercise #13

x=1 \sqrt{x}=1

X=? X=?

Video Solution

Step-by-Step Solution

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

  • Step 1: Square both sides of the equation.
  • Step 2: Simplify the resulting expression.

Now, let's work through each step:

Step 1: We have the equation x=1 \sqrt{x} = 1 .
Square both sides:

(x)2=12 (\sqrt{x})^2 = 1^2

Step 2: Simplify both sides of the equation.

The left side simplifies to x x , since the square and the square root cancel each other out:

x=1 x = 1

The right side simplifies to 1, so we have:

x=1 x = 1

Therefore, the solution to the problem is x=1 x = 1 .

Answer

1

Exercise #14

36= \sqrt{36}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Understand the definition of a square root.
  • Step 2: Identify which integer, when squared, gives 36.
  • Step 3: Verify this integer meets the required condition.
  • Step 4: Choose the correct answer from the given choices.

Now, let's work through each step:
Step 1: A square root of a number is a value that, when multiplied by itself, gives the original number. Here, we want y y such that y2=36 y^2 = 36 .
Step 2: We test integer values to find which one squared equals 36. Testing y=1,2,3,4,5, y = 1, 2, 3, 4, 5, and 6 6 gives:
- 12=1 1^2 = 1
- 22=4 2^2 = 4
- 32=9 3^2 = 9
- 42=16 4^2 = 16
- 52=25 5^2 = 25
- 62=36 6^2 = 36

Step 3: The integer 6 6 satisfies 62=36 6^2 = 36 . Therefore, 36=6 \sqrt{36} = 6 .

Step 4: The correct choice from the given answer choices is 6 (Choice 4).

Hence, the square root of 36 is 6 \mathbf{6} .

Answer

6

Exercise #15

x=2 \sqrt{x}=2

Video Solution

Step-by-Step Solution

To solve the problem, follow these steps:

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

Now, let's proceed through each step:
Step 1: The given equation is x=2\sqrt{x} = 2.
Step 2: Square both sides: (x)2=22(\sqrt{x})^2 = 2^2.
Step 3: This simplifies to x=4x = 4.

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

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

Answer

4