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Chapter 8: Problem 42

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

Short Answer

Expert verified
The evaluated expression is 0.125

Step by step solution

01

Substituting the values into the expression

Substitute the given values of \( x \) and \( t \) into the expression. This gives \( (1-0.5)^{3} \)
02

Simplifying the expression

First, simplify the expression inside the parentheses. This leads to \( 0.5^{3} \)
03

Calculating the power

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Write your answer as a power or as a product of powers. $$ 4 x \cdot\left(x \cdot x^{3}\right)^{2} $$

Solve the inequality. Then sketch a graph of the solution on a number line. $$|5+x|+4 \leq 11$$

Write your answer as a power or as a product of powers. $$ (-x)^{4}(-x)^{3}(-x)^{2} $$

Use substitution to solve the system. $$\begin{aligned}&-3 x+y=4\\\&-9 x+5 y=10\end{aligned}$$

Complete the statement using \(>\) or \(<\). $$\left(3^{6} \cdot 3^{12}\right) \geq 3^{72}$$
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