Question

When \(a+b=6,\) what is the value of \(2(a+b)+\frac{a+b}{6}+(a+b)^{2}-2 ?\) \(\mathbf{F} .23\) G. 37 H. 38 J. 43 K. 47

Short answer

Answer: K.47

Step-by-step solution

Substitute the given value

Since we are given that \(a+b=6\), we will substitute this value into the expression. So, the expression becomes: $$2(6)+\frac{6}{6}+(6)^{2}-2$$

Simplify and calculate the expression

Now, simplify the expression and perform the calculations: $$2(6)+\frac{6}{6}+(6)^{2}-2 = 12+1+36-2 = 47$$

Choose the correct answer

The value of the expression is 47 which corresponds to option K. Thus, the correct answer is \(\boxed{\text{K}.47}\).

Key concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition and multiplication. In the ACT Math Problem presented, the expression given is somewhat complex because it involves various operations incorporating the sum of two variables, denoted as 'a' and 'b'.

An important part of algebra is understanding how to work with these expressions, especially when given conditions like 'a + b = 6'. In such cases, you identify the variable part of the expression, which is 'a + b' and you 'plug in' the known value to eliminate the variables. Algebraic expressions often need to be manipulated or simplified to solve for unknowns or to plug in known values to find a specific numerical outcome.

Substitution Method

The substitution method is a crucial algebraic technique used to solve equations and evaluate expressions. For our problem, the substitution method involves taking the known value of the expression 'a + b' and replacing it with '6' wherever 'a + b' appears in the original problem. It's like swapping out a piece of a puzzle for the exact match, which then lets you see the full picture—or in our case, calculate the full value of the expression.

By substituting '6' for 'a + b', you simplify the problem to a numerical expression that no longer contains variables. Substitution is particularly useful for evaluating expressions and solving systems of equations. It's a method that requires careful attention to ensure that the substitution is applied consistently throughout the entire expression.

Simplifying Expressions

Once you've substituted the variables with numerical values, the next step is simplifying expressions. This key concept involves performing all the indicated operations according to the order of operations (parentheses, exponents, multiplication and division, and addition and subtraction).

In the ACT Math Problem, simplifying the expression means calculating the products, division, exponent, and sums in the correct order. First, we deal with the parentheses by multiplying and dividing as needed. Then, we tackle the exponent, and, finally, we pull the entire expression together by performing addition and subtraction. Simplifying helps to 'clean up' the expression, making it easier to grasp and solve. It is a sequential process that transforms a complex expression into a simpler, more understandable form.