Order of Operations: Roots - Examples, Exercises and Solutions

To solve the expression $6+\sqrt{64}-4=$, we need to follow the order of operations (PEMDAS/BODMAS):

• P: Parentheses (or Brackets)
• E: Exponents (or Orders, i.e., powers and roots, etc.)
• MD: Multiplication and Division (left-to-right)
• AS: Addition and Subtraction (left-to-right)

In this expression, we first need to evaluate the square root since it falls under the exponent category:

$\sqrt{64} = 8$

Next, we substitute the computed value back into the expression:

$6+8-4$

We then perform the addition and subtraction from left to right:

$6+8 = 14$

$14-4 = 10$

Thus, the final answer is:

$10$

Let's recall the order of operations:

1. Parentheses

2. Exponents and Roots

3. Multiplication and Division

4. Addition and Subtraction

There are no parentheses in this problem, so we'll start with exponents:

3*3+3² =

3*3+9 =

Let's continue to the next step, multiplication operations:

3*3+9 =

9 + 9 =

Now we're left with just a simple addition problem:

9+9= 18

And that's the solution!

Let's solve this problem step by step using the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction):

1. First, let's focus on what's inside the parentheses: $\sqrt{36}+9$

2. We need to evaluate the square root first:

• $\sqrt{36} = 6$ (because $6 \times 6 = 36$)

3. Now our expression looks like this: $2\times(6+9)$

4. Next, we perform the addition inside the parentheses:

• $6 + 9 = 15$

5. Our expression is now: $2\times15$

6. Finally, we perform the multiplication:

• $2 \times 15 = 30$

Therefore, $2\times(\sqrt{36}+9) = 30$

This matches the provided correct answer of 30.

In the first stage, let's calculate the powers of each of the terms:

$5^3=5\times5\times5=25\times5=125$

$5^2=5\times5=25$

$2^3=2\times2\times2=4\times2=8$

Now let's write the resulting expression:

$125:25\times8=$

Since the only operations in the expression are multiplication and division, we will solve the expression from left to right

In other words, we will divide first and then multiply:

$125:25=5$

$5\times8=40$

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.

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

So first calculate the values of the terms with exponents and then subtract the results:

$5^2\cdot4+3^3 =25\cdot4+27=100+27=127$Therefore, the correct answer is option B.

The given expression is $18^2-(100+\sqrt{9})$

We need to follow the order of operations (PEMDAS/BODMAS), which stands for:

• Parentheses
• Exponents (i.e., powers and square roots, etc.)
• MD Multiplication and Division (left-to-right)
• AS Addition and Subtraction (left-to-right)

Let's solve step by step:

Step 1: Evaluate the exponent and the square root in the expression:

• $18^2 = 324$
• $\sqrt{9} = 3$

So, the expression becomes $324 - (100 + 3)$

Step 2: Simplify the parentheses:

• 100 + 3 = 103 \)

So, the expression becomes $324 - 103$

Step 3: Subtract:

• 324 - 103 = 221 \)

Therefore, the value of the expression $18^2-(100+\sqrt{9})$ is 221.

We begin by solving the expression inside the parentheses $(20-3\times2^2)^2$. According to the order of operations (PEMDAS/BODMAS), we first handle any calculations inside parentheses and deal with exponents before performing multiplication, division, addition, or subtraction.

• Step 1: Solve the exponent inside the parentheses.

We have $2^2$, which equals $4$. Thus, the expression now is:



$(20-3\times4)^2$



• Step 2: Perform the multiplication inside the parentheses.

Multiply $3$ by $4$ to get $12$. The expression now simplifies to:



$(20-12)^2$



• Step 3: Perform the subtraction inside the parentheses.

Subtract $12$ from $20$. We get $8$. The expression now simplifies to:



$(8)^2$



• Step 4: Finally, solve the remaining exponent.

$8^2$ equals $64$.



Thus, $(20-3\times2^2)^2$ equals $64$.

We simplify each term according to the order from left to right:

$\sqrt{4}=2$

$4^2=4\times4=16$

$5^2=5\times5=25$$$

$\sqrt{1}=1$

Now we rearrange the exercise accordingly:

$2\times16-25\times1$

Since there are two multiplication operations in the exercise, according to the order of operations we start with them and then subtract.

We put the two multiplication exercises in parentheses to avoid confusion during the solution, and solve from left to right:

$(2\times16)-(25\times1)=32-25=7$

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

So first calculate the values of the terms with exponents and then subtract the results:

$(5-2)^2-2^3 =3^2-2^3=9-8=1$Therefore, the correct answer is option C.

According to the order of operations, we should first solve the expression inside of the parentheses:

$(\sqrt{380.25}-\frac{1}{2})=(19.5-\frac{1}{2})=(19)$

In the next step, we will proceed to solve the exponentiation, and finally the subtraction:

$(19)^2-11=(19\times19)-11=361-11=350$

To solve the expression $[(4-2)^2]^3$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Step 1: Solve the innermost parentheses:

The expression inside the innermost parentheses is $4-2$. We perform the subtraction:

$4-2 = 2$

Step 2: Apply the exponentiation:

Next, we take the result of the subtraction and apply the squaring operation $((2)^2)$:

$2^2 = 4$

Step 3: Apply the outer exponentiation:

Finally, we take the result of the previous step and raise it to the power of 3:

$4^3 = 64$

Therefore, the value of the expression $[(4-2)^2]^3$ is $64$.

$$This simple example illustrates the order of operations, which states that multiplication and division take precedence over addition and subtraction, and that operations within parentheses come first,

Let's say we have a fraction and a whole number (every whole number) between which a division operation takes place, meaning - we can relate to the fraction and the whole number as fractions in their simplest form, through which a division operation occurs, thus we can write the given fraction in the following form:

$\frac{(2^2-3)^{15}+4^2}{15+2}-\frac{3^2-2^2}{5}= \\ \downarrow\\ \big((2^2-3)^{15}+4^2\big):(15+2)-(3^2-2^2):5$We emphasize this by stating that we should relate to the fractions that are in the numerator and those in the denominator separately, as if they exist in their simplest form,

Let's return to the original fraction in question, meaning - in its given form, and simplify it, simplifying separately the different fractions that are in the numerator and those in the denominator (if simplification is needed), this is done in accordance with the order of operations mentioned above and in a systematic way,

We start with the first numerator from the left in the given fraction, noting that in this case it changes the fraction in the denominators that are in multiplication, therefore, we start with this fraction, this in accordance with the aforementioned order of operations, noting further that in this fraction in the denominators (which are in multiplication of 15) there exists a multiplication, therefore, we start calculating its numerical value in multiplication and then perform the subtraction operation that is in the denominators:

$\frac{(2^2-3)^{15}+4^2}{15+2}-\frac{3^2-2^2}{5}= \\ \frac{(4-3)^{15}+4^2}{15+2}-\frac{3^2-2^2}{5}= \\ \frac{1^{15}+4^2}{15+2}-\frac{3^2-2^2}{5} \\$ We continue with the fraction we received in the previous step and simplify the numerators and the denominators in the fraction, this is done in accordance with the order of operations mentioned above, therefore, we start calculating their numerical values in multiplication and then perform the division and subtraction operations that are in the numerators and in the denominators:

$\frac{1^{15}+4^2}{15+2}-\frac{3^2-2^2}{5}= \\ \frac{1+16}{17}-\frac{9-4}{5}= \\ \frac{17}{17}-\frac{5}{5}\\$ We continue and simplify the fraction we received in the previous step, again, in accordance with the order of operations mentioned above, therefore, we perform the division operation of the denominators, this is done systematically, and then perform the subtraction operation:

$\frac{17}{17}-\frac{5}{5}=\\ \frac{\not{17}}{\not{17}}-\frac{\not{5}}{\not{5}}=\\ 1-1=\\ 0$

We conclude with this, the steps of simplifying the given fraction, we received that:

$\frac{(2^2-3)^{15}+4^2}{15+2}-\frac{3^2-2^2}{5}= \\ \frac{1^{15}+4^2}{15+2}-\frac{3^2-2^2}{5} =\\ \frac{1+16}{17}-\frac{9-4}{5}= \\ 0$Therefore, the correct answer is answer D.