Problem 91 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$(7 t)^{3} \text { when } t=-\frac{3}{7}$$

Step-by-Step Solution

Answer

The value of the expression \((7 t)^3\) when \( t = -\frac{3}{7} \) is -1.

Step 1: Substitute the given value into the expression

First, replace the letter \( t \) in the expression \((7 t)^3\) with the value given for this variable, which is \(-\frac{3}{7}\). We do this by replacing every occurrence of \( t \) in the equation with \(-\frac{3}{7}\). This leads to the following equation: \((7(-\frac{3}{7}))^3\)

Step 2: Simplify inside the parentheses before the exponent

Next, use multiplication to combine the 7 and -3/7. This simplification gives us: \((-1)^3\)

Step 3: Raise the result to the power

Now, raise the result in parentheses to the third power. Because the base is -1, \((-1)^3\) simplifies to -1.

Key Concepts

Substitution

Substitution is the process of replacing a variable in an equation or an expression with a given value. This is a crucial step in evaluating expressions, as it allows us to find the numerical value of the expression based on specific inputs.

In our example, we are given the expression \((7t)^3\) and the value for \(t\) is \(-\frac{3}{7}\). To substitute, every instance of the variable \(t\) in the expression is replaced with \(-\frac{3}{7}\).

This transformation converts the original expression into \((7(-\frac{3}{7}))^3\). Successful substitution will set the stage for further operations and is essential for moving forward with simplification.

Simplifying expressions

Simplifying expressions involves performing all possible operations to reduce the expression into a simpler form. This often means combining like terms or carrying out basic arithmetic.

In our example of \((7(-\frac{3}{7}))^3\), simplification occurs inside the parentheses. The expression \(7(-\frac{3}{7})\) is simplified by multiplying. The 7 and \(-\frac{3}{7}\) are combined, as multiplying 7 by \(-\frac{3}{7}\) simplifies to -3.

As this simplification occurs, the fraction \(\frac{3}{7}\) cancels with the 7 outside, resulting in the expression \((-1)^3\). Thus, simplification reduces complexity, making subsequent calculations straightforward.

Multiplication of fractions

Multiplication of fractions is a foundational mathematical operation involving fractions. It is key when dealing with expressions like \(7(-\frac{3}{7})\).

When multiplying fractions, we multiply the numerators together and the denominators together. However, in our expression, 7 is actually a whole number, which can be thought of as \(\frac{7}{1}\).

To multiply 7 by \(-\frac{3}{7}\), we perform the following steps:

• Multiply the numerators: 7 times -3 gives -21.
• Multiply the denominators: 1 times 7 gives 7.
• Result: \(\frac{-21}{7}\), which simplifies to -3.

The simplification to -1 considering our exerting entire term results by interpreting that -3 as the ultimate -1 achieved when raised to the power of 3. Understanding this process is critical for simplifying and evaluating expressions correctly.