Question

Evaluate the expression for the given value of the variable. (Review 1.2) $$6 y^{5} \text { when } y=4$$

Short answer

The solution to evaluating the expression \(6y^5\) when 'y' equals 4 is \(6144\).

Step-by-step solution

Substitution

To start off, substitute the given value for 'y' into the expression. This means replacing 'y' in '6y^5' with 4: \(6 * 4^5\)

Evaluating the exponent

Next, calculate the value of the exponent \(4^5\). This equals 1024.

Multiply

Then multiply the result of the exponent (1024) by 6: \(6 * 1024 = 6144\).

Key concepts

Exponentiation

Exponentiation is a form of mathematical notation indicating the number of times a number (the base) is multiplied by itself. The exponent tells us how many times to multiply the base by itself. For example, in the expression 6y^5, y is the base, and 5 is the exponent.

Calculating exponentiation requires us to multiply the base, y, by itself 5 times. If y equals 4, as in the exercise, then 4^5 means we calculate 4 * 4 * 4 * 4 * 4. When we compute this, we get 1024. It's important to understand exponentiation because it occurs frequently in algebra, and mastering it will help you simplify and evaluate expressions accurately.

Here's a small tip: When you have small bases and exponents, you could multiply them step by step, but for larger numbers, using a calculator would speed up the process and help ensure accuracy.

Substitution in Algebra

Substitution in algebra is a technique used to replace a variable in an expression with its actual value. This method is essential because it allows us to evaluate algebraic expressions to find their numerical value. In the given exercise, we use substitution to replace 'y' with the number 4 in the expression 6y^5.

This substitution gives us a new expression: 6 * 4^5. Once the substitution is made, we can proceed with other operations like exponentiation and multiplication to evaluate the expression. Substitution is not limited to numbers; you can also substitute one expression for a variable in another expression. However, it's crucial to ensure that the substitution is appropriate for the expression/problem context.

To improve clarity, remember to always write the new expression clearly after substituting, before moving on to the next step.

Arithmetic Operations

Arithmetic operations encompass the basic mathematical operations of addition, subtraction, multiplication, and division. They are the building blocks of more complex mathematical concepts. In the context of evaluating algebraic expressions, once we perform the substitution, we use these operations to find the value of the expression.

In the problem given, after substitifying 4 for y and determining the value of 4^5, we are left with the operation 6 * 1024, which is a multiplication. Carrying out this multiplication gives us the final answer, 6144. While multiplication may seem straightforward, it's important to pay close attention to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to accurately evaluate expressions.

Practicing these operations regularly helps to improve speed and accuracy, which is particularly beneficial when dealing with lengthy or complex algebraic expressions.