Powers and Roots - Basic - Examples, Exercises and Solutions

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

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

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

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.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's start with converting the square root to an exponent using the law mentioned in a':

$\sqrt{x^2}= \\ \downarrow\\ (x^2)^{\frac{1}{2}}=$We'll continue using the law of exponents mentioned in b' and perform the exponent operation on the term in parentheses:

$(x^2)^{\frac{1}{2}}= \\ x^{2\cdot\frac{1}{2}}\\ x^1=\\ \boxed{x}$Therefore, the correct answer is answer 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 expression $5+\sqrt{36}-1=$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Here are the steps:

First, calculate the square root:

$\sqrt{36} = 6$

Substitute the square root back into the expression:

$5 + 6 - 1$

Next, perform the addition and subtraction from left to right:

Add 5 and 6:

$5 + 6 = 11$

Then subtract 1:

$11 - 1 = 10$

Finally, you obtain the solution:

$10$

The root of 441 is 21.

$21\times21=$

$21\times20+21=$

$420+21=441$

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.

To solve the expression $143-\sqrt{121}+18$, we need to follow the order of operations, which dictate that we should simplify any expressions under a square root first, followed by subtraction and addition.

Step 1: Simplify the square root:

• Calculate the square root: $\sqrt{121}$.
• The square root of 121 is 11, because $11 \times 11 = 121$.

Now, substitute back into the expression:

• The expression becomes: $143 - 11 + 18$.

Step 2: Perform the subtraction:

• Calculate $143 - 11$.
• This equals 132, because subtracting 11 from 143 yields 132.

Step 3: Perform the addition:

• Now add 18 to the result of the subtraction: $132 + 18$.
• The result is 150, because adding 18 to 132 equals 150.

Therefore, the final answer is $150$.

To solve the expression $81+\sqrt{81}+10$, we need to follow the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Here, the expression contains a square root, which is a type of exponent operation. Therefore, we will handle the square root first:

• Find the square root of 81, which is calculated as follows: $\sqrt{81} = 9$.

Now substitute the result back into the original expression:

$81 + 9 + 10$

Next, perform the addition operations from left to right:

• First, add 81 and 9: $81 + 9 = 90$.
• Then, add the result to 10: $90 + 10 = 100$.

Therefore, the final result of the expression $81+\sqrt{81}+10$ is:

$100$

To solve the expression $4^2 - 4^3$, we start by evaluating each power separately:

• Calculate $4^2$:
  $4^2$ means $4$ multiplied by itself, which is $4 \times 4 = 16$.

• Calculate$4^3$:
  $4^3$ means $4$ multiplied by itself three times, which is $4 \times 4 \times 4 = 64$.

Next, substitute these values back into the expression:

• $4^2 - 4^3 = 16 - 64$

Perform the subtraction:

• $16 - 64 = -48$

Thus, the correct answer is $-48$.

To solve the exercise $5^2-4^1=$, we need to follow the order of operations, specifically focusing on powers (exponents) before performing subtraction.

• Step 1: Calculate $5^2$. This means we multiply 5 by itself: $5 \times 5 = 25$.

• Step 2: Calculate $4^1$. Any number to the power of 1 is itself, so $4^1 = 4$.

• Step 3: Subtract the result of $4^1$ from $5^2$: $25 - 4$.

• Step 4: Complete the subtraction: $25 - 4 = 21$.

Thus, the correct answer is $21$.

To solve the expression $5^2 - 4^2 + 2^2$, we'll need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Here, we only have exponents and basic arithmetic.

• First, calculate the powers:
  $5^2 = 25$,
  $4^2 = 16$,
  $2^2 = 4$.


• Now substitute the calculated values back into the expression:
  $25 - 16 + 4$.


• Perform the subtraction and addition from left to right:
  $25 - 16 = 9$.


• Then add 4 to 9:
  $9 + 4 = 13$.

The final answer is $13$.

To solve the expression $6^3 - 6^2$, we will follow the order of operations, which in this case involves evaluating the powers before the subtraction operation.

• First, evaluate $6^3$:
  • $6^3$ means $6 \times 6 \times 6$.
  • Calculating this, we get $6 \times 6 = 36$.
  • Then multiply 36 by 6 to get $36 \times 6 = 216$.
• Next, evaluate $6^2$:
  • $6^2$ means $6 \times 6$.
  • Calculating this gives us $36$.
• Finally, subtract the second result from the first:
  • That is $216 - 36$.
  • Performing the subtraction, we get $180$.

Thus, the result of the expression $6^3 - 6^2$ is $180$.

To solve the expression $7^1 - 7^2$, we need to evaluate the powers first before performing the subtraction. The steps are as follows:

• Calculate $7^1$: Since any number to the power of 1 is the number itself, we have $7^1 = 7$.
• Calculate $7^2$: This means 7 is multiplied by itself, which gives us $7 \times 7 = 49$.
• Subtract the results: Now, perform the subtraction $7 - 49$.
• This yields: $7 - 49 = -42$.

Thus, the correct answer is $-42$.

To solve the expression $4\times\sqrt{0.49}+4^2 =$, we will follow the order of operations, which prioritizes operations inside parentheses, exponents and roots, followed by multiplication and division, and finally addition and subtraction.

1. Calculate the Square Root:
The first step is to solve the square root part of the expression. $\sqrt{0.49}$.
$0.49$ is a simple decimal number whose square root is $0.7$, because $0.7 \times 0.7 = 0.49$.
So,$\sqrt{0.49} = 0.7$.

2. Multiply:
Next, we multiply the result of the square root by 4:
$4 \times 0.7 = 2.8$.

3. Calculate the Power:
Evaluate $4^2$.
$4^2 = 16$, because $4 \times 4 = 16$.

4. Addition:
Now, add the results from the previous steps:
$2.8 + 16 = 18.8$.

The final result of the expression $4\times\sqrt{0.49}+4^2$ is $\boxed{18.8}$.