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Understanding how to evaluate algebraic expressions is key to mastering algebra. Evaluating an expression involves substituting numbers for variables and then performing the arithmetic operations as indicated in the expression.

Take for example the expression \( (4 n^{3})^{2} \). Before you can evaluate it, you must know the value of \( n \). As seen in the exercise solution, after solving the equation \( 2 n - 3 = 4 n - 7 \), you find that \( n = 2 \). Substituting \( n \) back into the expression, you then follow the operation order to compute the final value.

Remember, for accurate evaluation, pay close attention to the order of operations and carry them out correctly. Often this will involve working with exponents, multiplication, and brackets —sometimes all in one expression like in the given problem.

When it comes to solving algebraic expressions, the BIDMAS rule, also known as PEMDAS in some regions, is the order of operations used to solve them accurately. BIDMAS stands for Brackets, Indices (Powers and Roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

In practice, this rule means you calculate parts of a mathematical expression inside brackets first, followed by any exponents (indices). Next, you handle any divisions or multiplications, and finally any additions or subtractions.

For instance, in the expression \( (4 \times n^3)^2 \), because of BIDMAS, you would first compute the exponent \( n^3 \) before multiplying by 4. After that, you would square the result. By following BIDMAS, you avoid common mistakes that could lead to the wrong answer.

Substitution is a fundamental concept in algebra where you replace a variable with a number or another expression. It's a powerful technique that allows you to solve equations and evaluate expressions.

After solving \( 2 n - 3 = 4 n - 7 \) and finding \( n = 2 \), you perform substitution by replacing every instance of \( n \) with 2 in the original expression \( (4 n^3)^2 \). You then follow the order of operations to simplify and find the value of the expression.

Why is substitution important?

• It helps solve equations by finding the value of unknown variables.
• It allows you to evaluate expressions for specific values.
• It's an essential step in simplifying expressions and functions.

Mastering substitution will enable you to tackle a wide range of algebraic challenges, making it a key skill in your math toolkit.