Question

Evaluate the expression for the given value of the variable. $$2\left(y^{3}\right) \text { when } y=-5$$

Short answer

The evaluation of the expression \(2y^{3}\) when \(y=-5\) is \(-250\).

Step-by-step solution

Identify the expression and given value

The expression given is \(2y^{3}\) and the value of \(y\) is \(-5\).

Substitute the value into the expression

To evaluate the expression, replace \(y\) in the expression with the given value. That gives us \(2(-5)^{3}\).

Solve the exponent first

Following the order of operations (PEMDAS/BODMAS), solve the exponent first. Hence, compute \((-5)^{3}\), which is \(-125\).

Multiply the result by 2

Multiply the result from step 3 by 2, which gives \(2* -125 = -250\).

Key concepts

Order of Operations

When it comes to evaluating algebraic expressions, it is crucial to follow the order of operations. This ensures that all mathematicians get the same result when evaluating an expression. The common acronym for remembering this order is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In some countries, BODMAS is used, standing for Brackets, Orders (another word for exponents), Division, Multiplication, Addition, and Subtraction.

Using this guideline, an expression like \(2(y^3)\) is simplified by addressing the operations in the correct order. First, we deal with the parentheses and any numbers or expressions inside them, then any exponents are resolved, followed by multiplication and division in the order they appear from left to right, and finally, addition and subtraction the same way. Applying the rules of PEMDAS/BODMAS to our example, the exponent \(y^3\) is dealt with before we multiply the result by 2.

Substitution Method

The substitution method is a way to evaluate expressions by replacing variables with their given values. This is particularly useful in algebra when you're working with expressions that include one or more variables, and you need to find out what the expression equates to with specific values.

In our exercise example \(2(y^3)\), we are told that \(y = -5\). To evaluate the expression, we replace each 'y' in the expression with '-5'. It becomes \(2(-5)^3\). Substitution makes it easier to see what the next steps are. After the substitution, we proceed with the order of operations to simplify and find the value of the expression.

Exponents

In algebra, an exponent denotes how many times a number, known as the base, is multiplied by itself. The exponent is the small number written above and to the right of the base number. For example, in \(y^3\), 3 is the exponent indicating that \(y\) is multiplied by itself three times: \(y \times y \times y\).

It's important to note that negative bases can affect the sign of the answer. In the calculation of \((-5)^3\), the negative sign is raised to an odd power, which will result in a negative outcome. If our exponent was even, such as \((-5)^2\), the result would be positive because the negative sign would be multiplied an even number of times, cancelling itself out. Exponents are a powerful tool in algebra, fundamentally important in operations ranging from basic arithmetic to complex equations.