Problem 42 - Algebra 1

Question

Evaluate the expression. $$(1-x)^{t} \text { when } x=0.5 \text { and } t=3$$

Step-by-Step Solution

Answer

The evaluated expression is 0.125

Step 1: Substituting the values into the expression

Substitute the given values of \( x \) and \( t \) into the expression. This gives \( (1-0.5)^{3} \)

Step 2: Simplifying the expression

First, simplify the expression inside the parentheses. This leads to \( 0.5^{3} \)

Step 3: Calculating the power

Perform the calculation \( 0.5^{3} \). This results in 0.125

Key Concepts

Substitution in Algebra

Understanding substitution in algebra is a crucial skill for evaluating expressions. To substitute is to replace a variable with a specific value.
For instance, if you have the expression \(1-x)^t\), and you know that \(x=0.5\) and \(t=3\), you substitute by plugging in these numbers as follows: \(1-0.5)^3\). This process turns an algebraic expression into a simpler or more concrete numerical expression that you can evaluate.
Make sure to perform the substitution carefully to avoid mistakes, especially if the expression has more than one variable or if it involves negative numbers. Substitution forms the basis of solving equations and understanding more complex topics in algebra.

Exponents and Powers

Exponents and powers are shorthand for telling us how many times to multiply a number by itself. The exponent is the small number found above and to the right of the base number.
For example, in the expression \(0.5^3\), the base is 0.5 and the exponent is 3, which means \(0.5\) multiplied by itself 3 times: \(0.5 \times 0.5 \times 0.5\). The result, \(0.125\), represents the volume of a cube with sides of \(0.5\) units for this particular exercise.
Understanding how to work with exponents is essential for simplifying expressions and working through problems in algebra. Remember the basic rules, such as any number to the power of 0 is 1 and to the power of 1 is the number itself.

Simplifying Algebraic Expressions

Simplifying algebraic expressions makes them easier to work with. The goal is to transform complex or lengthy expressions into their simplest form.
For the expression \( (1-x)^3\) after substituting \(x=0.5\), we get \(0.5^3\). To simplify this, perform the exponentiation, which involves multiplying \(0.5\) by itself three times. The simplified result for our example is \(0.125\).
While simplifying, always check if there are like terms to combine, rules of exponents to apply, or parentheses that can be eliminated. Simplifying correctly is key to revealing the most basic form of an expression, ultimately making it easier to solve or manipulate in algebraic operations.