Problem 19 - Elementary and Intermediate Algebra

Question

Write each expression in an equivalent form using an exponent. $$ 4 t \cdot 4 t \cdot 4 t \cdot 4 t $$

Step-by-Step Solution

Answer

The expression can be written as \(4^4t^4\).

Step 1: Recognizing the Expression

The expression given is: \(4t \cdot 4t \cdot 4t \cdot 4t\). Notice that the expression consists of identical factors being multiplied.

Step 2: Grouping Base Terms

Observe that each factor of the expression is composed of a number, \(4\), and a variable, \(t\). We need to separate the number and the variable for easier handling.

Step 3: Writing the Numbers

Acknowledging that there are four '4's in the expression, we can write them as \(4^4\).

Step 4: Writing the Variables

Similarly, there are four 't's in the expression, which can also be written using an exponent, \(t^4\).

Step 5: Combining the Number and Variable

Now, put the two parts together to express the original multiplication with exponents: \(4^4t^4\). This represents the original expression using exponents.

Key Concepts

Multiplication of Variables

When dealing with algebra, multiplying variables can often seem a bit tricky at first, but it's all about recognizing patterns. In algebra, when you multiply the same variable several times, you're really dealing with repeated multiplication. This is where exponents come into play.

Instead of writing each instance of the variable, you can simplify the expression by using an exponent. The number of times the variable is repeated becomes the exponent. For example, if you have four 't's as in our original expression, you can express this as \(t^4\). The exponent here tells you how many 't's are being multiplied together, saving space and making expressions easier to read and understand.

**Key Points:**

• Multiplying a variable is repeated multiplication.
• Use exponents to simplify repeated multiplications.
• \(t \cdot t \cdot t \cdot t = t^4\)

Understanding how multiplication of variables works with exponents is a stepping stone to mastering algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). In our exercise, the expression is made up of the number 4 and the variable \(t\), expressed as \(4t \cdot 4t \cdot 4t \cdot 4t\). This is a multilineal expression involving both numeric and variable components.

Algebraic expressions like this are common in algebra. They're used to model real-world problems and help solve equations. Recognizing common patterns like repeated multiplication allows for simplification using mathematical rules like those governing exponents.

**Simplifying Algebraic Expressions:**

• Identify similar terms or repeated operations.
• Apply mathematical rules like those for exponents to simplify.
• Maintain balance by treating numbers and variables consistently.

By understanding and simplifying algebraic expressions, math problems become more manageable, making way for more complex topics.

Use of Exponents in Algebra

Exponents play a crucial role in algebra by simplifying expressions that involve repeated multiplication. When you see multiple instances of the same term being multiplied together, you can use an exponent to denote this repetition. For example, \(4t \cdot 4t \cdot 4t \cdot 4t\) simplifies to \(4^4t^4\).

Here are some basics about using exponents in algebra:

**Exponents Basics:**

• The base is the number or variable being multiplied.
• The exponent tells you how many times to multiply the base by itself.
• \(a^n\) means 'a' is multiplied 'n' times by itself.

The power of exponents becomes especially valuable when dealing with large expressions in equations or when simplifying expressions to solve problems. Practicing the use of exponents not only enhances your algebra skills but also builds a foundation for more advanced math topics.