Question

$$a^{3}+10 a \text { when } a=3$$

Short answer

57

Step-by-step solution

Substitute the value of a

Replace \(a\) in the equation \(a^{3}+10a\) with \(3\). This gives us \(3^{3} + 10*3\).

Carry out the operations

Carry out the operations: \(3^{3} = 27\) and \(10*3 = 30\).

Add the results

Add the results from Step 2 together: \(27 + 30 = 57\).

Key concepts

Algebraic Expressions

At the heart of algebra lies the concept of algebraic expressions. These are combinations of numbers, variables (such as 'a', 'b', or 'x'), and arithmetic operations like addition, subtraction, multiplication, and division. More complex expressions might include exponents, roots, and functions.

Take, for example, the expression \(a^3 + 10a\). Here, 'a' represents a variable that can take on any value. The term \(a^3\) means that 'a' is raised to the power of three, while the term \(10a\) signifies ten times the value of 'a'. Algebraic expressions are versatile tools that allow us to describe patterns, construct equations, and solve problems in a wide range of mathematical scenarios.

Evaluation of Expressions

Evaluation of expressions is a fundamental skill in algebra that involves replacing variables with numbers to calculate the numerical value of the expression. When we evaluate an expression, we must follow the correct order of operations often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

Considering the exercise \(a^3 + 10a\) when \(a=3\), the first step is to substitute the variable 'a' with the given number. This modification transforms the expression into a series of operational steps, leading to a concrete result. Careful and methodical replacement of variables with actual values is crucial for accurate evaluation.

Exponents

Exponents are an efficient way to denote repeated multiplication of a number by itself. An exponent is written as a superscript to the right of the base number. For instance, \(3^3\) is read as 'three raised to the third power' and signifies \(3 \times 3 \times 3\).

In the exercise, the term \(3^3\) represents the cube of 3, which is 27. Understanding how to work with exponents allows us to simplify and resolve algebraic expressions systematically. Remember: a positive integer exponent tells us how many times to multiply the base by itself. The bigger the exponent, the larger the number grows, conveying exponential growth.