Powers and Roots: Using parentheses

First, calculate the square root: $\sqrt{25} = 5$.

Then, add the result to 7: $5 + 7 = 12$.

Finally, multiply by 3: $3 \times 12 = 36$.

Let's solve the expression given in the problem: $(\sqrt{16}-2^2+6):2^2=$

We need to follow the Order of Operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform calculations in the correct order:

• $\sqrt{16}$ simplifies to $4$ because the square root of 16 is 4.
• Next, evaluate $2^2$, which equals $4$.
• Replace these results back into the expression: $(4-4+6):4$.

Now let's go step-by-step through the simplified expression:

• Calculate inside the parentheses: $4 - 4 + 6$.
• $4-4$ equals $0$.
• Then, $0+6$ equals $6$.
• We now have $6:4$, which denotes division.
• Perform the division: $6 \div 4 = 1.5$.

Thus, the solution to the expression is: $1.5$.

Let's solve the expression $(2+1\times2)^2$ step-by-step, adhering to 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)).

Firstly, handle the expression inside the parentheses $(2+1\times2)$:

• Within the parentheses, according to PEMDAS, we first perform the multiplication $1\times2$ which equals $2$.
• Now, the expression inside the parentheses becomes $(2+2)$.
• Next, perform the addition: $2+2=4$.

Now the expression simplifies to $4^2$.

Second, handle the exponent:

• Calculate the square of 4: $4^2 = 16$.

Thus, the final answer is $16$.

Let's solve the expression step by step using the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right)).

The expression is: $(15+9:3-4^2)^2$

• Start by solving the expression inside the parentheses: $15+9:3-4^2$

• Handle the division first: $9:3 = 3$

• The expression now becomes $(15+3-4^2)$

• Next, handle the exponent: $4^2 = 16$

• The expression now becomes $(15+3-16)$

• Finally, perform the addition and subtraction from left to right:
  $15+3 = 18$
  $18-16 = 2$

• Now, we need to take this result and square it: $2^2 = 4$

Thus, the final solution to the problem $(15+9:3-4^2)^2$ is: $4$

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.

Let's walk through the steps to solve the expression $(4^2:8):2+3^2$ using the correct 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)).

• First, resolve the expression inside the parentheses: $4^2:8$

  • The exponent comes first:

    $4^2 = 16$, so the expression now is $16:8$.

• Next, perform the division inside the parentheses: $16:8$ equals 2. So the expression within the parentheses simplifies to 2.

• Now, we replace the original expression with this simplified result:

  $2:2+3^2$

• We perform the division: $2:2 = 1$.

• Substitute back into the expression:

  $1+3^2$

• Next, calculate the exponent:

  $3^2 = 9$.

• Finally, add the results:

  $1 + 9 = 10$.

Thus, the solution to the expression $(4^2:8):2+3^2$ is 10.

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.

Previously mentioned in the order of arithmetic operations, exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the root terms inside the parentheses first) :

$(\sqrt{100}-\sqrt{9})^2:7 = (10-3)^2:7 =7^2:7=\frac{7^2}{7}$where in the second step we simplified the expression in parentheses, and in the next step we wrote the division operation as a fraction,

Next we'll calculate the value of the numerator by performing the exponentiation, and in the next step we'll perform the division (essentially reducing the fraction):

$\frac{7^2}{7} =\frac{\not{49}}{\not{7}}=7$Therefore the correct answer is answer A.

Previously mentioned in the order of arithmetic operations, exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the terms with exponents inside the parentheses first) :

$5:(13^2-12^2) =5:(169-144) =5:25=\frac{5}{25}$where in the second step we simplified the expression in parentheses, and in the next step we wrote the division operation as a fraction,

Then we'll perform the division (we'll actually reduce the fraction):

$\frac{\not{5}}{\not{25}}=\frac{1}{5}$Therefore the correct answer is answer C.

Previously mentioned in the order of operations that exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the terms inside the parentheses first) :

$(10^2-2\cdot5):3^2 = (100-10):3^2 =90:3^2=\frac{90}{3^2}$where in the second stage we simplified the expression in parentheses, and in the next stage we wrote the division operation as a fraction,

Next we'll calculate the value of the term in the fraction's numerator by performing the exponent, and in the next stage we'll perform the division (essentially reducing the fraction):

$\frac{90}{3^2} =\frac{\not{90}}{\not{9}}=10$Therefore the correct answer is answer D.

Previously mentioned in the order of operations that exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate then first the value of the expression inside the parentheses (by calculating the roots inside the parentheses first) :

$(\sqrt{9}-\sqrt{4})^2\cdot4^2-5^1 =(3-2)^2\cdot4^2-5^1 =1^2\cdot4^2-5^1$where in the second stage we simplified the expression in parentheses,

Next we'll calculate the values of the terms with exponents:

$1^2\cdot4^2-5^1 =1\cdot16-5$then we'll calculate the results of the multiplications

$1\cdot16-5 =16-5$and after that we'll perform the subtraction:

$16-5=11$Therefore the correct answer is answer B.

To solve the expression $(18-10)^2+3^3$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Step 1: Parentheses
  First, solve the expression inside the parentheses: $18-10$.
  $18-10 = 8$

• Step 2: Exponents
  Next, apply the exponents to the numbers:
  $(8)^2$ and $3^3$.
  $8^2 = 64$
  $3^3 = 27$

• Step 3: Addition
  Finally, add the results of the exponentiations:
  $64 + 27$
  $64 + 27 = 91$

Thus, the final answer is $91$.

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.

To solve $(\frac{3}{4})^2$, you need to square both the numerator and the denominator separately:

1. Square the numerator: $3^2 = 9$

2. Square the denominator: $4^2 = 16$

3. Combine the results to get $\frac{9}{16}$

To solve $(\frac{5}{6})^2$, you need to square both the numerator and the denominator separately:

1. Square the numerator: $5^2 = 25$

2. Square the denominator: $6^2 = 36$

3. Combine the results to get $\frac{25}{36}$

To solve the expression $2\times(3^3+\sqrt{144})$, we 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)).

• Step 1: Begin by solving the part inside the parentheses $(3^3+\sqrt{144})$.

• Exponents: Calculate $3^3$.

  $3^3$ means $3 \times 3 \times 3 = 27$.

• Roots: Calculate $\sqrt{144}$.

  $\sqrt{144} = 12$, since 12 is the number that when multiplied by itself gives 144.

• Add the results of the exponent and the root: $27 + 12 = 39$.

• Step 2: Multiply the sum by 2:

  $2 \times 39 = 78$.

Therefore, the final answer is $78$.

To solve the expression $(3+1)^2-(4+1)$, we need to apply the order of operations, which is also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right)).

Step 1: Simplify inside the parentheses.
Start with the expression inside the first parentheses: $3+1$. Adding the numbers gives:
$3+1 = 4$.

Step 2: Apply the exponent.
Next, we take the result from step 1 and apply the exponent:
$4^2$. This equals:
$4 \times 4 = 16$.

Step 3: Simplify inside other parentheses.
Now, simplify the expression inside the second set of parentheses: $4+1$. This gives:
$4+1 = 5$.

Step 4: Perform subtraction.
Finally, subtract the result of the second parentheses from the exponent result:
$16 - 5$, which equals:
$11$.

Thus, the final result of the expression $(3+1)^2-(4+1)$ is 11.

To solve the expression $3-(5^2:5)^2+7^2$, we should follow the order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Here are the steps to solve the expression:

1. Evaluate the exponents

• Calculate $5^2$ which equals $25$.

• Calculate $7^2$ which equals $49$.

2. Evaluate expressions inside parentheses

• The expression inside the parentheses is $5^2:5$ which simplifies to $25:5 = 5$.

3. Evaluate the expression inside the parentheses raised to a power

• The simplified expression now is $(5)^2$, which is $25$.

4. Substitute back into the expression

• The original expression now becomes: $3 - 25 + 49$.

5. Perform the addition and subtraction from left to right

• First, calculate $3 - 25$ which equals $-22$.

• Then, $-22 + 49$ equals $27$.

Therefore, the final result of the expression $3-(5^2:5)^2+7^2$ is $27$.

To solve the expression $(4^2+3)\times\sqrt{9}=$, we need to follow the order of operations, also known as PEMDAS/BODMAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• First, we deal with the exponent $4^2$, which means 4 raised to the power of 2. Calculate $4^2 = 16$.

• Next, we calculate the expression inside the parentheses: $(4^2+3)$. We already know $4^2 = 16$, so we add 3 to 16: $16 + 3 = 19$.

• Then, we find the square root of 9: $\sqrt{9} = 3$.

• Lastly, we perform the multiplication: $19 \times 3 = 57$.

Thus, the final result of the expression $(4^2+3)\times\sqrt{9}$ is 57.

To solve the given power expression, we need to apply the formula for powers of a fraction. The expression we are given is:
$\left(\frac{2}{3}\right)^3$

Let's break down the steps:

• When we raise a fraction to a power, we apply the exponent to both the numerator and the denominator separately. This means raising both 2 and 3 to the power of 3.
• Thus, we calculate:
  $2^3 = 8$ and $3^3 = 27$.
• Therefore, $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$.

So, the result of the expression $\left(\frac{2}{3}\right)^3$ is $\frac{8}{27}$.