Problem 31 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for then given value of the variable. \(h^{5}\) when \(h=2\)

Step-by-Step Solution

Answer

The value of the expression \(h^{5}\) when \(h=2\) is 32.

Step 1: Substitute the Value

Replace \(h\) in \(h^{5}\) with the given value. Substituting \(h = 2\) into the expression gives us \(2^{5}\).

Step 2: Evaluate Expression

The operation here is to raise 2 to the power of 5. Start by multiplying 2 by itself, and repeat this for a total of 5 times. So, \(2^{5} = 2 * 2 * 2 * 2 * 2\).

Step 3: Calculate Final Result

Now we calculate the multiplication, giving us a final result of 32. So, \(h^{5}\) when \(h=2\) equals to 32.

Key Concepts

Evaluate Expression

Evaluating an expression involves performing operations to simplify or find the value of a given mathematical phrase. In the case of our exercise, the expression is in the form of an exponent, specifically, a power. To evaluate the expression \( h^{5} \) where \( h = 2 \), it's essential to perform the given operation—raising the base \( h \) to the power of 5.

To evaluate the expression correctly, you generally want to ensure all parts of the expression are ready for calculation. First, substitute any variables, and then systematically follow the mathematical operations specified, either by order of operations or through step-by-step simplification as seen with exponents.

The process in this exercise is straightforward once you replace the variable with the given number, turning the abstract expression into an arithmetic operation that can be solved with careful calculation.

Substitution Method

The substitution method is a method used in mathematics to replace variables in an expression or equation with specific values. This technique can be invaluable for solving or simplifying equations, particularly when an equation contains one or more variables. In our example, the variable \( h \) is given the value 2, allowing us to substitute and simplify \( h^{5} \) into \( 2^{5} \).

Here’s how the substitution method works in simple steps:

• Identify the variable you need to replace.
• Replace each instance of the variable with the given number.
• After substitution, perform the operations as instructed by the new expression.

By substituting \( h = 2 \), you transform a variable-heavy expression into a concrete number operation that brings you closer to the solution. It changes the focus from abstract algebra to concrete arithmetic, making it simpler to find the final answer.

Powers of Numbers

Understanding powers of numbers is crucial in mathematics, especially when dealing with exponents. In this exercise, \( 2^{5} \) represents the operation of multiplying the number 2 by itself multiple times—five times, in this case.

When you see an expression like \( a^b \), it means the base \( a \) is multiplied by itself \( b \) times. So for \( 2^{5} \), you calculate it by performing the multiplication: \( 2 \times 2 \times 2 \times 2 \times 2 \).

This concept highlights some key points about powers:

• The base is the number that you multiply.
• The exponent tells you how many times the base is used as a factor.
• Multiplying repeatedly gives the expression value—here, calculated as 32.

Understanding powers makes handling large numbers through multiplication less cumbersome, helping students grasp the effects of exponential growth and the efficiency of repeated multiplication.