Q55. - Algebra 2

Question

Write expressions having values from one to ten using exactly four 4s. You may use any combination of the operation symbols $+,-,\times ,÷,$ and/or grouping symbols, but no other numbers are allowed. An example of such an expression with a value of zero is $\left(4+4\right)-\left(4+4\right)$

Step-by-Step Solution

Answer

$\left(44-4\right)÷4$Expression having values one to 10 using four 4s are

• Expression having value 1 is$\left(4+4\right)÷\left(4+4\right)$
• Expression having value 2 is $\left(4÷4\right)+\left(4÷4\right)$
• Expression having value 3 is $\left[\left(4\times 4\right)-4\right]÷4$
• Expression having value 4 is $\left[\left(4-4\right)\times 4\right]+4$
• Expression having value 5 is $\left[\left(4\times 4\right)+4\right]÷4$
• Expression having value 6 is $\left[\left(4+4\right)÷4\right]+4$
• Expression having value 7 is $\left(4+4\right)-\left(4÷4\right)$
• Expression having value 8 is $\left[\left(4\times 4\right)-4\right]-4$
• Expression having value 9 is $\left[\left(4÷4\right)+4\right]+4$
• Expression having value 10 is $\left(44-4\right)÷4$

Step 1 - Order of simplification

To evaluate any expression involving more than one operation, follow the below order of operations:

• First evaluate expressions inside grouping symbols like parentheses ( ) or brackets [ ].
• Evaluate the exponents.
• Perform Multiplication and division from left to right.
• Perform addition and subtraction from left to right.

Step 2 - Write expression having value 1

Let the expression be $\left(4+4\right)÷\left(4+4\right)$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis and then simplify as shown below

$\begin{array}{c}\left(4+4\right)÷\left(4+4\right)=8÷8\\ =1\end{array}$

Thus the expression having value 1 is $\left(4+4\right)÷\left(4+4\right)$

Step 3 - Write expression having value 2

Let the expression be$\left(4÷4\right)+\left(4÷4\right)$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis and then simplify as shown below

$\begin{array}{c}\left(4÷4\right)+\left(4÷4\right)=1+1\\ =2\end{array}$

Thus the expression having value 2 is $\left(4÷4\right)+\left(4÷4\right)$

Step 4 - Write expression having value 3

Let the expression be $\left[\left(4\times 4\right)-4\right]÷4$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis then simplify the term inside bracket and finally simplify further as shown below

$\begin{array}{c}\left[\left(4\times 4\right)-4\right]÷4=\left[16-4\right]÷4\\ =12÷4\\ =3\end{array}$

Thus the expression having value 3 is $\left[\left(4\times 4\right)-4\right]÷4$

Step 5 - Write expression having value 4

Let the expression be $\left[\left(4-4\right)\times 4\right]+4$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis then simplify the term inside bracket and finally simplify further as shown below

$\begin{array}{c}\left[\left(4-4\right)\times 4\right]+4=\left[0\times 4\right]+4\\ =0+4\\ =4\end{array}$

Thus the expression having value 4 is $\left[\left(4-4\right)\times 4\right]+4$

Step 6 - Write expression having value 5

Let the expression be $\left[\left(4\times 4\right)+4\right]÷4$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis then simplify the term inside bracket and finally simplify further as shown below

$\begin{array}{c}\left[\left(4\times 4\right)+4\right]÷4=\left[16+4\right]÷4\\ =20÷4\\ =5\end{array}$

Thus the expression having value 5 is $\left[\left(4\times 4\right)+4\right]÷4$

Step 7 - Write expression having value 6

Let the expression be $\left[\left(4+4\right)÷4\right]+4$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis then simplify the term inside bracket and finally simplify further as shown below

$\begin{array}{c}\left[\left(4+4\right)÷4\right]+4=\left[8÷4\right]+4\\ =2+4\\ =6\end{array}$

Thus the expression having value 6 is $\left[\left(4+4\right)÷4\right]+4$$\left[\left(4+4\right)÷4\right]+4$

Step 8 - Write expression having value 7

Let the expression be $\left(4+4\right)-\left(4÷4\right)$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis and then simplify as shown below

$\begin{array}{c}\left(4+4\right)-\left(4÷4\right)=8-1\\ =7\end{array}$

Thus the expression having value 7 is $\left(4+4\right)-\left(4÷4\right)$

Step 9 - Write expression having value 8

Let the expression be $\left[\left(4\times 4\right)-4\right]-4$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis then simplify the term inside bracket and finally simplify further as shown below

$\begin{array}{c}\left[\left(4\times 4\right)-4\right]-4=\left[16-4\right]-4\\ =12-4\\ =8\end{array}$

Thus the expression having value 8 is $\left[\left(4\times 4\right)-4\right]-4$

Step 10 - Write expression having value 9

Let the expression be $\left[\left(4÷4\right)+4\right]+4$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis then simplify the term inside bracket and finally simplify further as shown below

$\begin{array}{c}\left[\left(4÷4\right)+4\right]+4=\left[1+4\right]+4\\ =5+4\\ =9\end{array}$

Thus the expression having value 9 is $\left[\left(4÷4\right)+4\right]+4$

Step 11 - Write expression having value 10

Let the expression be $\left(44-4\right)÷4$

To evaluate the value of the this expression, first simplify the expression inside the parenthesis and then simplify as shown below

$\begin{array}{c}\left(44-4\right)÷4=40÷4\\ =10\end{array}$

Thus the expression having value 10 is $\left(44-4\right)÷4$