Write the expression as a square of a monomial:
$121 a^6$
Answer (1)
Find the square root of the coefficient: 121 = 11 .
Determine the exponent of the monomial: a 6 = ( a 3 ) 2 , so the exponent is 3.
Write the expression as a square of a monomial: ( 11 a 3 ) 2 .
The expression 121 a 6 can be written as ( 11 a 3 ) 2 .
Explanation
Understanding the Problem We are asked to express 121 a 6 as a square of a monomial. This means we want to find an expression of the form ( c a n ) 2 such that ( c a n ) 2 = 121 a 6 , where c is a constant and n is an integer.
Finding the Constant First, we need to find a number c such that c 2 = 121 . The square root of 121 is 11, so c = 11 since 1 1 2 = 121 .
Finding the Exponent Next, we need to find an integer n such that ( a n ) 2 = a 6 . Using the properties of exponents, we know that ( a n ) 2 = a 2 n . Therefore, we need to find n such that 2 n = 6 . Dividing both sides by 2, we get n = 3 .
Writing the Expression Now we can write the expression as ( 11 a 3 ) 2 . Let's check if this is correct: ( 11 a 3 ) 2 = 1 1 2 × ( a 3 ) 2 = 121 × a 3 × 2 = 121 a 6 . This matches the original expression.
Examples
Imagine you are designing a square garden with an area of 121 a 6 square feet. To build a fence around the garden, you need to know the length of each side. By expressing the area as a square of a monomial, ( 11 a 3 ) 2 , you find that each side of the garden is 11 a 3 feet long. This helps you determine the amount of fencing material needed.
