Problem 45 - Algebra 1: Concepts and Skills

Question

USING EXPONENTS Evaluate the expression. $$ 12+y^{3} \text { when } y=3 $$

Step-by-Step Solution

Answer

The evaluated value of the expression is 39.

Step 1: Understand the given expression

The given expression to evaluate is \(12+y^{3}\). We are also given that \(y=3\)

Step 2: Substitute the value of y

Substitute the value of \(y = 3\) into the expression, resulting in \(12+3^{3}\)

Step 3: Calculate the expression

Here, \(3^{3}\) is 27 as \(3*3*3 = 27\). Adding 12 to 27 equals 39.

Key Concepts

Using Exponents

When you come across numbers raised to a power, like in the expression \(12+y^3\), you are working with exponents. An exponent, often referred to as a power, tells us how many times to multiply a number by itself. For example, \(y^3\) means multiplying \(y\) by itself three times: \(y \times y \times y\).

In algebra, evaluating expressions with exponents requires two crucial steps. First, identify the base — which in this case is \(y\) — and the exponent, here it's 3. Second, perform the exponentiation before any other operations. Remember that order matters. According to standard mathematical conventions, exponentiation precedes addition and subtraction in the order of operations.

Substitution in Algebra

Substitution is like a game of swapping. In the expression \(12+y^3\), you're told that \(y=3\).

To evaluate this expression, you'll want to play the substitution game: everywhere you see a \(y\), replace it with 3. This swapping makes the abstract concrete, turning an expression with a variable into one with only numbers. In our equation, substitution transforms \(12+y^3\) into \(12+3^3\), which is much easier to handle because it involves familiar arithmetic. Substitution is a fundamental tool in algebra that helps simplify expressions and solve equations. It's like telling a story with characters' names, but then you find out who each character really is — the story becomes much clearer!

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It tells us how many times to use the base as a factor in a multiplication. To excel at exponentiation, recognize these key points:

• It's a form of repeated multiplication.
• The base is the number that is repeatedly multiplied.
• The exponent, written as a small number above and to the right of the base, indicates the number of times to multiply the base by itself.
• The result of exponentiation is called a power.

For instance, in \(3^3\), we multiply 3 by itself three times, resulting in \(3 \times 3 \times 3 = 27\). This process simplifies seemingly complex expressions by reducing them to single numbers, making subsequent calculations more manageable. Understanding exponentiation is essential for higher-level math, such as algebra, calculus, and beyond.