Multiply. Simplify if possible.
a) $(8+\sqrt{13})^2=$ $\square$
b) $(-8+2 \sqrt{13})^2=$ $\square$
Note: To input $a \sqrt{b}$ type "a sqrt(b)". To input $a \sqrt[b]{c}$ type "a root(b)(c)".
Answer (1)
Expand ( 8 + 13 ) 2 using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 to get 64 + 16 13 + 13 .
Simplify the expression to 77 + 16 13 .
Expand ( − 8 + 2 13 ) 2 using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 to get 64 − 32 13 + 52 .
Simplify the expression to 116 − 32 13 .
The final answers are 77 + 16 13 and 116 − 32 13 .
Explanation
Understanding the Problem We are asked to multiply and simplify two expressions involving square roots.
a) ( 8 + 13 ) 2 =
b) ( − 8 + 2 13 ) 2 =
We will use the formula ( a + b ) 2 = a 2 + 2 ab + b 2 to expand and simplify each expression.
Expanding the first expression a) Expanding ( 8 + 13 ) 2 :
Using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 , where a = 8 and b = 13 , we have:
( 8 + 13 ) 2 = 8 2 + 2 ( 8 ) ( 13 ) + ( 13 ) 2
= 64 + 16 13 + 13
= 77 + 16 13
Expanding the second expression b) Expanding ( − 8 + 2 13 ) 2 :
Using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 , where a = − 8 and b = 2 13 , we have:
( − 8 + 2 13 ) 2 = ( − 8 ) 2 + 2 ( − 8 ) ( 2 13 ) + ( 2 13 ) 2
= 64 − 32 13 + 4 ( 13 )
= 64 − 32 13 + 52
= 116 − 32 13
Final Answer Therefore, the simplified expressions are:
a) ( 8 + 13 ) 2 = 77 + 16 13
b) ( − 8 + 2 13 ) 2 = 116 − 32 13
Examples
Understanding how to expand and simplify expressions like these is useful in various areas of mathematics, such as algebra and calculus. For example, when solving quadratic equations or finding the distance between two points, you might encounter expressions involving square roots that need to be simplified. Imagine you are building a square garden with sides of length ( 8 + 13 ) meters. To calculate the area of the garden, you would need to square this expression, which is exactly what we did in part (a). The simplified form, 77 + 16 13 , gives you a more manageable way to understand the garden's area.
