Evaluate the Function g(-2)=6(0.5)^-2

Solution:

Step 1:

Insert the value $-2$ into the function $g(x)$ to get $g(-2)=6(0.5{)}^{-2}$.

Step 2:

Proceed to simplify the expression.

Step 2.1:

Apply the negative exponent rule, which states $b^{-n}=\frac{1}{b^n}$, to transform the expression into $g(-2)=6\left(\frac{1}{(0.5{)}^2}\right)$.

Step 2.2:

Calculate $(0.5{)}^2$ to get $g(-2)=6\left(\frac{1}{0.25}\right)$.

Step 2.3:

Simplify the fraction by finding the reciprocal of $0.25$.

Step 2.3.1:

Multiply $6$ by the reciprocal of $0.25$, which is $4$, to get $g(-2)=6\times 4$.

Step 2.3.2:

Simplify the multiplication to get $g(-2)=24$.

Step 2.4:

Conclude that the final result is $24$.

Step 3:

There is no further action required as the solution is complete.

Knowledge Notes:

To solve the given problem, several mathematical concepts and rules are applied:

1. Function Evaluation: This involves substituting a specific value into a function to find the output. In this case, $-2$ is substituted into $g(x)$.

2. Negative Exponents: The rule for negative exponents states that $b^{-n}=\frac{1}{b^n}$. This is used to transform terms with negative exponents into their reciprocal form.

3. Exponentiation: Calculating $b^n$ when $b$ is a fraction involves multiplying the base $b$ by itself $n$ times. For example, $(0.5{)}^2=0.5\times 0.5=0.25$.

4. Multiplication and Division: Multiplying or dividing numbers is a basic arithmetic operation. In this case, $6$ is multiplied by the reciprocal of $0.25$ to simplify the expression.

5. Simplification: This is the process of rewriting an expression in a simpler or more concise form without changing its value. The simplification steps involve applying arithmetic operations and algebraic rules to reduce the expression to its simplest form.

By applying these concepts, the problem is solved systematically, leading to the final answer of $24$.