Basis of a power - Examples, Exercises and Solutions

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 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 $4^2 - 4^3$, 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 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 $6^3 - 6^2$, 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 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 $3-(5^2:5)^2+7^2$, 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 $27$.

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.

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 $91$.

To solve the expression $2^4-4:2^2$, we must follow the order of operations, also known as BIDMAS/BODMAS (Brackets, Indices/Orders, Division/Multiplication, Addition/Subtraction).

$1.$ Start with calculating the powers (indices) in the expression:

• $2^4 = 16$
• $2^2 = 4$

$2.$ Substitute these values back into the expression:

$16 - 4 : 4$

$3.$ Next, perform the division:

$4 : 4 = 1$

$4.$ Substitute back again and perform the final subtraction:

$16 - 1 = 15$

Therefore, the solution to the expression $2^4-4:2^2$ is 15.

To solve the expression $4^2:2+5^2$, we need to follow the order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This guide will help us apply the correct sequence to solve the problem.

• Step 1: Exponents
  First, we solve the exponents in the expression. In this case, we have $4^2$ and $5^2$.
  Calculate each:
  $4^2 = 4 \times 4 = 16$
  $5^2 = 5 \times 5 = 25$

• Step 2: Division
  Next, we perform the division operation. In the expression $16 : 2$, divide 16 by 2:
  $16 : 2 = 8$

• Step 3: Addition
  Finally, we add the results from the previous steps together:
  $8 + 25 = 33$

Thus, the value of the expression $4^2:2+5^2$ is $33$.

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 $\frac{8}{27}$.

Using the formula for the area of a square whose side is b:

$S=b^2$In the picture, we are presented with three squares whose sides from left to right have a length of 6, 3, and 4 respectively:

Therefore the areas are:

$S_1=3^2,\hspace{4pt}S_2=6^2,\hspace{4pt}S_3=4^2$square units respectively,

Consequently the total area of the shape, composed of the three squares, is as follows:

$S_{\text{total}}=S_1+S_2+S_3=3^2+6^2+4^2$square units

To conclude, we recognise through the rules of substitution and addition that the correct answer is answer C.