Problem 46 - Algebra 1: Concepts and Skills

Question

USING EXPONENTS Evaluate the expression. $$ x^{5}+10 \text { when } x=1.5 $$

Step-by-Step Solution

Answer

The evaluation of the expression \(x^{5}+10\) when \(x=1.5\) is \(17.59375\)

Step 1: Identify the value for x

From the problem, it's apparent that \(x = 1.5\)

Step 2: Substitute x in the expression

Place the value of \(x\) which is \(1.5\) into the expression. This results in \(1.5^{5} +10\)

Step 3: Calculate \(x^{5}\)

Compute the value of \(1.5^{5}\) using a calculator or a basic knowledge of exponents. This results in \(7.59375\}

Step 4: Add the calculated value to 10

This step gives \(7.59375 + 10 = 17.59375\)

Key Concepts

Evaluation of Expressions

When dealing with algebraic expressions, evaluating means finding the value of an expression by performing the operations indicated. In algebraic expressions that include exponents, such as the one given in our problem, evaluations involve several steps. Here's the general process:

• Identify the expression to evaluate (e.g., \(x^5 + 10\)).
• Find out which values you need, usually given in the problem statement (e.g., \(x = 1.5\)).
• Follow the necessary order of operations, also known as BIDMAS/BODMAS (Brackets, Indices/Orders, Division, Multiplication, Addition, Subtraction).

Breaking down calculations in this manner helps ensure accuracy and makes it easier to tackle more complex expressions.

Substitution

Substitution is one of the cornerstones of solving algebraic expressions. It involves replacing a variable with a given numerical value. This is especially useful in simplifying or evaluating expressions. In our example:

• The task is to substitute \(x\) with \(1.5\) in the expression \(x^5 + 10\).
• This results in replacing the \(x\) with \(1.5\) where it appears, forming the new expression \(1.5^5 + 10\).

Substitution allows us to simplify the expression and prepare it for further calculations, like evaluating the exponents.

Basic Arithmetic Operations

Basic arithmetic operations are fundamental in mathematics and are used to carry out day-to-day calculations. These operations include addition, subtraction, multiplication, division, and working with exponents.Let's break down the steps we followed:

• Start with exponents: Compute \(1.5^5\). Using a calculator or mathematical tools, we find \(1.5^5 = 7.59375\). In this step, we are applying the power or exponent operation.
• Next, proceed to addition: Add the exponent result \(7.59375\) to \(10\), leading to \(7.59375 + 10 = 17.59375\).

Understanding and applying these basic operations correctly is vital for accurate computation in algebra.