Problem 78 - Algebra 1

Question

Evaluate the expression. \(t^{4}-t\) when \(t=7\)

Step-by-Step Solution

Answer

The evaluated expression \(t^{4}-t\) when \(t=7\) is 2394.

Step 1: Substitution of Variable

Substitute \(t=7\) in the equation \(t^{4}-t\). This gives \(7^{4}-7\).

Step 2: Power Calculation

Calculate \(7^{4}\) separately before subtracting 7. This equals 2401.

Step 3: Subtraction

Now subtract 7 from the result of Step 2. So, \(2401-7 = 2394\)

Key Concepts

Substitution in Algebra

One of the fundamental techniques in algebra is the concept of substitution. Substitution involves replacing a variable in an algebraic expression with its numerical value. This method is essential because it transforms the algebraic expression into a form that can be processed using basic arithmetic operations. For example, in the exercise we're discussing, the variable t is given a value of 7.

When the substitution step is carried out, each instance of t in the expression is replaced with 7, thus changing the expression from t to the power of 4 minus t into 7 to the power of 4 minus 7. By doing so, the algebraic expression becomes a numerical expression that can be evaluated using standard arithmetic. It is crucial to perform the substitution carefully to avoid any errors in the subsequent steps.

Exponentiation

Exponentiation is the process of raising a number to a power, which is noted by an exponent. The number being raised to a power is called the base, and the exponent indicates how many times the base is multiplied by itself. In our example, 7^{4} represents 7 multiplied by itself 4 times: 7 × 7 × 7 × 7.

Calculating Powers

Understanding and accurately calculating powers is crucial in algebra. For many students, exponentiation can be daunting, especially with larger exponents. However, mastering this concept is possible with practice and by breaking it down into smaller multiplications. After substituting 7 for t, we calculate 7^{4} separately to ensure precision in our work, which gives us 2401.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value when the variables are replaced with actual numbers. An expression can be as simple as a single variable, like x, or as complex as a multifaceted formula involving multiple operations and variables. The beauty of an algebraic expression lies in its versatility; it can model real-world situations and solve problems systematically.

In context with our exercise, t^{4}-t is an algebraic expression with variable t. Once we substitute t with 7 and work through the operations—exponentiation followed by subtraction—we arrive at the expression's value. The subtraction is the final step, completing the transformation from an abstract expression to a concrete numerical value.